The first one is the infamous Speed Warrior:
During the Battle Phase of the turn you Normal Summon this card, you can double its original ATK until the end of the Battle Phase.
Speed Warrior has an Ignition Effect. Oddly enough, this effect is activated during the Battle Phase instead of the Main Phases. It's a very strange exception. Other than that, the effect itself is very simple. It takes the Original ATK of Speed Warrior, doubles that value, and sets that value as the Current ATK. In other words, it works much like Shrink (*thunder*), only that it doubles the value instead of halving it.
Then, we have Max Warrior:
When this card attacks an opponent's monster, it gains 400 ATK during damage calculation only. If this card destroys an opponent's monster by battle, this card's Level becomes 2 and its original ATK and DEF are halved until your next Standby Phase.
Gaining ATK is a Continuous Effect. The effect that weakens Max Warrior, on the other hand, is a Mandatory Trigger Effect. It activates at the End of the Damage Step. Max Warrior actually does modify its Original ATK/DEF, instead of setting a new Current ATK value. If multiple monsters are destroyed by Max Warrior, then the ATK/DEF is halved once again for each monster destroyed.
Moving on, we have Shield Warrior:
During damage calculation, you can remove from play this card from your Graveyard to prevent monsters you control from being destroyed by this battle.
Shield Warrior has an Optional Quick Effect that you activate at the "Damage Calculation" Sub-Step. Removing it from Play is a cost. Probably the most common question about its effect is how long it lasts, as its wording is somewhat ambiguous. The key term here is the end of the sentence: "by this battle". This means that Shield Warrior only prevents the destruction of your monsters in battle from this single attack.
Another member of the Warriors is Turret Warrior:
You can Tribute 1 Warrior-Type monster to Special Summon this card from your hand. If you do, it gains ATK equal to the Tributed monster's original ATK.
Turret Warrior only has a Summoning Condition. Note that it's not restrictive in any way: You can still Normal Summon it, Set it, or Special Summon it with other Card Effects. The Inherent Special Summon itself is self-explanatory. If Turret Warrior has its effects negated (Skill Drain), the gained ATK disappears.
Speaking of Inherent Special Summons, we have Level Warrior:
If there are no monsters on the field, you can Normal Summon this card from your hand as a Level 2 monster. If your opponent controls a monster and you control no monsters, you can Special Summon this card from your hand as a Level 4 monster.
Both abilities are Summoning Conditions. However, Level Warrior's Level is modified in the Hand, not on the Field. This means that it will not reset to Level 3 if Level Warrior has its effects negated. Not much can be said, other than the Level 4 Summoning Condition is an Inherent Special Summon.
One more Warrior is Gaunlet Warrior:
You can Tribute this card to have all face-up Warrior-Type monsters you control gain 500 ATK and DEF until the end of the next Damage Step they battle. This effect can be activated during either player's turn.
Gaunlet Warrior has an optional Quick Effect. It can be activated during the Damage Step, but not at the "Damage Calculation" Sub-Step. Tributing Gaunlet Warrior is a cost, and this affect boots the ATK of the Warriors that are face-up when the effect resolves (not those Summoned afterwards). The effect lasts until the End of the Damage Step in which monster battles, meaning it entered the Battle Step.
And finally, we have the new Warrior monster: Rapid Warrior:
During your Main Phase 1, you can activate this card's effect. If you do, it can attack your opponent directly this turn. Other monsters cannot attack during the turn your activate this effect.
Rapid Warrior has an Ignition Effect. It allows it to attack directly. Even if Skill Drain is activated later, it's still allowed to attack directly. Other than that, the effect is very simple.
That covers another share of Yusei's Cards. If you have any questions, feel free to drop me an e-mail at ness00[at]gmail[dot]com.