The easiest Summoning Condition to explain are the restrictions on Normal Summons or Sets. They have no other implications other than what the text say: The only way to Summon the monster is through a Special Summon, or a Flip Summon. This restriction is mainly used to force you to fulfill the rest of the Summoning Condition, such as preventing you to Tribute Summon Judgment Dragon if you don't run Lightsworn monsters. Of course, in practice, this would be quite ridiculous. This restriction has some funny peculiar cases, such as the Mystic Swordsman LV4 and LV6, which cannot be Normal Summoned but must be Set instead, or Rare Metal Dragon, which can't be Normal Summoned or Set but offers no means to Special Summon itself. What a selfish guy, no wonder Dragons aren't welcome at parties.
Then we have the restrictions and benefits of Special Summons. These are very simple in the rare cases in which a monster can be Normal Summoned, Set, and also has a Summoning Condition about its Special Summon. Whether it is Light and Darkness Dragon, which can be Normal Summoned, Set, and can't be Special Summoned, or a monster like Machina Fortress which can be placed on the Field by those three means, they don't bring up any major complications: When someone asks what you can and can't do with Light and Darkness Dragon, it's only restiction is that it can't be Special Summoned under any circumstances. And about Machina Fortress, you are free to do what you want. Often, monsters like Machina Fortress have individual rulings reminding you that "you can Normal Summon this monster", because, like we will see in a minute, it's the other guys who become a nuisance, but are also the most popular monsters.
When you have a monster that can't be Normal Summoned or Set, it'll most likely be accompanied by a brief description on how to Summon it. This can be a Monster Effect or an Inherent Special Summon. This small line also determines the future of the monster after it leaves the Field, explaining if other Effects will be able to Summon it or not.
There are two types of restrictions. The first is the type that reads that "this Card can only be Special Summoned...". This restriction means that the only way you can initially place this monster on the Field is through the following Inherent Special Summon or Effect. Despite how absolute it sounds, once the monster has been successfully Special Summoned by the described method, one can use any effects to place that monster back into the Field, such as Call of the Haunted or Return from the Different Dimension. Monsters with this restriction are called "Special Summon-only" monsters. It shouldn't be to hard to remember since those three words are present on their text. Since a Special Summon-only monster needs to be Summoned first by the method described in its text, you cannot use other means to Summon it. The usual example is to send the monster from the Deck to the Graveyard or to discard it, and then try to use Monster Reborn on it. Since you didn't perform the requirements of the monster, you cannot use Monster Reborn on it. However, if you do fulfill the requirements, and the monster is later destroyed, it is perfectly legal to use Monster Reborn. Do note that you do not need to fulfill the conditions again while using an effect to Summon the monster. For example, you do not need to Remove from Play 3 Fiend-type monsters to Special Summon Dark Necrofear while using Monster Reborn.
It is worth mentioning that the Special Summon must be successful. If this Summon is negated by Solemn Judgment or Royal Oppression, the monster will not be considered to be Summoned properly, and so, it cannot be Summoned from the Graveyard or Removed from Play Zone. As an additional note, if the monster returns to the Deck or the Hand after being Summoned properly, it will no longer be recognized as "that" monster (as it is lost in a location that isn't public knowledge), so you have to Summon it properly once again. And as a lame note, if the Special Summon-only monster is flipped face-down after meeting its requirements, and while trying to Flip Summon it, this Summon is negated, it will not be considered to be Summoned properly, giving the same results as negating its Special Summon: One isn't able to Special Summon it by other Card Effects. Some examples of Special Summon-only monsters:
This card cannot be Normal Summoned or Set. This card can only be Special Summoned by removing from play 1 LIGHT and 1 DARK monster in your Graveyard.
The Dark Creator:
This card cannot be Normal Summoned or Set. This card can only be Special Summoned, by having 5 or more DARK monsters in your Graveyard while you control no monsters.
This card cannot be Normal Summoned or Set. This card can only be Special Summoned by Tributing 2 monsters on your opponent's side of the field, and is Special Summoned to your opponent's side of the field.
The other type of restriction is the type that reads "this Card cannot be Special Summoned except ...". Unlike the previous one, this restriction IS as blunt as it sounds. Other than the described method, it is NOT possible to Special Summon the monster by any means, even if you have succeeded in Summoning it once. These monsters are usually called "Nomi" monsters. Nomi being a Japanese word meaning "only", as in, "the only possible way". And without being able to Summon these monsters afterwards, the text in their description certainly becomes the "only" way. Some people tend to call Special Summon-only monsters "Semi-Nomi", which causes some confusion. First, the similarity between both names. Second, how semi-nomi means "almost/half only", which doesn't seem to go either way. Is it the only way? Are the more? More than one? Who knows. Finally, while I think Nomi monsters were never given a specific name in English, Special Summon-only is an official term refferenced in a bunch of rulings. Just...try to avoid semi-nomi, for the sake of little cute animals of your choice. Specially mini-mooses.
Back to the monsters themselves, there actually is one elusive way to Special Summon these monsters that just don't want to get into the Field. That is, a Card that ignores their Summoning Conditions. Even in that case, Nomi monsters still have hefty requirements: If you try to Special Summon a Nomi monster from your Graveyard, you MUST still meet the same requirements than those of a Special Summon-only monster. So, for example, if you use Foolish Burial on Horus the Black Flame Dragon LV8, you cannot use Level Modulation on it. You must first Summon Horus LV8, and once it is sent to the Graveyard, you can use Level Modulation, ignoring its Summoning Conditions, and finally Special Summoning it back to the Field. With less than 10 Cards being able to ignore these Conditions, and taking so much effort, it rather seems you are better off playing by the rules than trying to find backdoors, isn't it? Some examples of Nomi monsters:
Dark Armed Dragon:
This card cannot be Normal Summoned or Set. This card cannot be Special Summoned except by having exactly 3 DARK monsters in your Graveyard.XX-Saber Faultroll:
This card cannot be Normal Summoned or Set. This card cannot be Special Summoned except while you control 2 or more face-up "X-Saber" monsters.
This card cannot be Normal Summoned or Set. This card cannot be Special Summoned except with "Commander Covington"
Mostly bringing up these three to show that the relevant phrase here is that they "cannot be Special Summoned". Many were puzzled about whether Faultroll was a Nomi or not because it didn't use the words "except by". We judges usually tell players to look for the words "except by" because "can" and "cannot" sound and look pretty much alike. However, what you should truly look at is that the Card itself tells you that there is no other way to Special Summon it. The same applies to Machina Force, which traded the older text of "except by the effect of..." for a shorter "except with".
A heavily asked question about Special Summon-only and Nomi monsters is where can they be Summoned from. Since they often do not specify a location, one can simply assume that they can be Summoned from anywhere as long as you fulfill their conditions. The sad truth is that, if no location is specified, the monster is Summoned from the Hand. Take a look at the six examples in this article, and you will see that none of them ever mentions being Summoned from the Hand. This rule becomes quite a grudge for judges who have been around for a while like myself (since 2004). Back in the day, everyone was familiar with this rule. Looking at older sources, I couldn't find any bright neon sign saying it, so I really don't know how we managed to figure this rule. As a quick poll, ask any veteran player about this, and they will definately agree. But as the player base shifts, and new players come in, this unwritten rule is easily ignored. So now go! Spread the word!
Wait, don't just go yet, we still need to address more Summons Q_Q I still owe you some comments regarding Fusion, Ritual, and Synchro Summons. These three apply to their corresponding Monster Cards (Purple, Blue, and White). They are all Special Summons, too.
Let's start with Fusion Monsters. A Fusion Summon is always performed through a Card Effect, be it Polymerization, Instant Fusion, Dragon's Mirror, etc. Sending Fusion Materials to the Graveyard for a Fusion Summon is never a cost. Fusion Monsters are placed in your Extra Deck, and they are Special Summon-only monsters by default. That is to say, once you Special Summon them properly through a Fusion Summon, you can later use Monster Reborn and similar Cards on them. However, if you use Gale Dogra to send them to the Graveyard, this is not possible. Despite being Special Summon-only monsters, there are some Cards that can Special Summon Fusion Monsters to the Field without Fusion Summoning them. Of course, this isn't considered to properly Summon the monster, so you cannot use Monster Reborn on the Fusion Monster afterwards. Some examples include Summoner of Illusions or Cyber Stein.
Some Fusion Monsters are also Nomis, like Dark Paladin or most Elemental Hero Fusions. Also, some Fusion Monsters are Special Summon-only monsters, but they are not Summoned through a Fusion Summon. In other words, you don't Summon the Fusion through a Fusion Summon, but rather, you Summon the Fusion through the Fusion's Summoning Conditions. Cheap puns are cheap. Back to a serious tone, monsters like the Gladiator Beast Fusion Monsters, or the XYZ Fusion Monsters do not need to and cannot be Special Summoned through a Fusion Summon. They can be identified by mentioning that "You do not use Polymerization" (although it covers any other similar Cards too. So, for example, you cannot choose to Summon them via Future Fusion. You also cannot use Cyber Stein on these Fusion Monsters which do not use Polymerization.
Some examples include:
Blue-Eyes Ultimate Dragon:
"Blue-Eyes White Dragon" + "Blue-Eyes White Dragon" + "Blue-Eyes White Dragon"
Elemental Hero Absolute Zero:
1 "Elemental Hero", "Destiny Hero", or "Evil Hero" Monster + 1 WATER monster
This monster cannot be Special Summoned except by Fusion Summon.
Chimeratech Fotress Dragon:
"Cyber Dragon" + 1 or more Machine-Type monsters
You can only Special Summon this card from your Extra Deck by sending the above cards from either side of the field to the Graveyard. (You do not use "Polymerization".)
In these examples, we see that Blue-Eyes Ultimate Dragon doesn't have any restrictions, so it's simply your average Fusion Monster. Absolute Zero, on the other hand, is a Nomi monster. Finally, Chimeratech Fotress Dragon is one of the Fusion Monsters that is not Summoned through a Fusion Summon, but rather, its own Inherent Special Summon.
Moving on, we have Ritual Monsters. These are Summoned by activating a Ritual Spell Card when you have the Ritual Monster in your Hand, offering the necessary Tributes demanded in the Ritual Spell Card. The Tributes in a Ritual Summon are not a cost. Ritual Monsters are also Special Summon-only monsters by default, so if you discard them, you cannot use Monster Reborn on them. Similar to Fusion Monsters, there are some Cards that can Special Summon Ritual Monsters from their default location (the Hand) without Ritual Summoning them. This is what Ritual Foregone does. Similar to Fusion Monsters, this does not properly Ritual Summon the monster, so once again, Monster Reborn cannot be used.
Finally, Synchro Monsters are the newest mechanic. In order to perform a Synchro Summon, you must control a Tuner Monster and other monsters described in the Synchro Monster's text, as well matching the Level of the Synchro Monster with those face-up monsters that it requires. You then send these monsters to the Graveyard to Synchro Summon the Synchro Monster. So, for example, you may use a Level 2 Tuner Monster and a Level 6 Non-Tuner monster to Special Summon a Level 8 Synchro Monster, if that's what it requires. Synchro Summons are performed as an Inherent Special Summon, so unlike Ritual or Fusion Summons, these can be negated by Solemn Judgment or Thunder King Rai-Oh. The requirements of Synchro Monsters vary a lot, asking for monsters of a certain name, Attribute, Type, etc.. Synchro Monsters are Special Summon-only monsters by default as well, but there are also some Nomi Synchro Monsters, such as Trident Dragion. Let's take a look at a few examples:
1 Tuner + 1 or more non-Tuner monsters
"Junk Synchron" + 1 or more non-Tuner monsters
1 Tuner + 2 or more non-Tuner monsters
1 Tuner + 1 or more non-Tuner Spellcaster-Type monsters
1 Tuner + 1 or more EARTH monsters
1 Dragon-Type Tuner + 1 or more non-Tuner Dragon-Type monsters
This monster cannot be Special Summoned except by Synchro Summon.
We see that Stardust requirements are any Tuner Monster and any number of Non-Tuner monsters. Junk Synchron has similar requirements, but the Tuner Monster MUST be named "Junk Synchron". Mist Wurm once again can use any Tuner and Non-Tuner monsters, but you cannot use only one Non-Tuner. Arcanite Magician is similar to Stardust Dragon, but all the Non-Tuner monsters must be Spellcaster-Type monsters. XX-Saber Gottoms also requires monsters with the EARTH Attribute, but they don't need to be Non-Tuners (which means you can use a bunch of EARTH Tuner monsters to Synchro Summon Gottoms). Finally, Trident Dragion requires a specific Type of Tuner Monster, a specific Type of Non-Tuner monsters, and it is also a Nomi monster.
Guess that covers up every form of Summoning a monster. If you have any questions, feel free to drop me an e-mail at ness00[at]gmail[dot]com.