Continuous Effects are the last type of Monster Effect. They aren't as surprising or as manageable as the other type of effects in a strategic sense, and some of their mechanics scare little children and haunt them in their nightmares. The shortest and probably the most global way to define a Continuous Effect is an effect that doesn't use the chain. Now that's awfully vague, isn't it? After reading a little longer, you'll see that this is the only thing they all have in common.
The most popular form of Continuous Effects are those that are applied all the time, as long as the monster that generates it remains face-up on the Field. An obvious example would be Jinzo's effect of preventing the activation of Trap Cards: You can't chain to it, it is applied all the time, and once Jinzo is no longer face-up, the effect stops. That's simple enough.
The problems start appearing once you take monsters whose Continuous Effect is only active some times. The Rulebook itself uses Mind Protector's effect as an example, quoting its effect that prevents monsters with 2000 ATK or less from attacking. This effect is similar enough to Jinzo's, in the aspect that it only works while Mind Protector is face-up on the Field. However, Protector is only applied when a monster whose ATK is low enough would try to declare an attack. That is to say, it doesn't need to be applied during the Draw Phase, or the opponent's End Phase. The same goes for monsters that attack directly, or monsters that can inflict piercing damage.
Moving on, some monsters have effects that are applied during other Card's resolutions or between chain links. Such blasphemy is possible thanks to Continuous Effects, which do not use the chain, and so, you aren't trying to "activate" an effect while a chain or an effect is resolving. The most common example for this situation are monsters that self-destruct themselves when you don't meet a condition, such as the Earthbound Immortals or Giant Kozaky. These effects will be applied between chain links.
Are you confused yet? Let's take it to the next level. A common example of monsters whose Continuous Effect is applied during other Card's resolution are monsters that can prevent another monster's destruction, or even their own, such as the various Six Samurai or Defender, the Magical Knight. As soon as the destructive effect tries to affect these monsters, you can apply their effects and prevent the destruction, sometimes even causing improper resolutions. For example, if Gemini Spark is used on a Six Samurai monster that prevents its destruction, then the player who activated Gemini Spark will not draw a Card.
As a somewhat obvious note, effects that are applied "all the time" are also applied during other Card's resolutions, such as Cyber Blader's third effect, that will negate effects at their resolution.
Weirdness piles up when Continuous Effects replace other effects, such as how Armed Sea Hunter's negation effect is a Trigger Effect, but Dark Ruler Ha Des' negation is actually Continuous. And just look at Makyura the Destructor, what on Earth is that and how is it Continuous? ARGH >_>
In short, Continuous Effects are extremely miscellaneous, and the only real way to point them out is if one can figure out that they do not use the chain. Individual rulings helps, and so does tradition (such as piercing effects). They can also be identified for wearing a hockey mask and a chainsaw.
And with that lame refference, that sums up Monster Effects. If you have any questions, feel free to drop me an e-mail at ness00[at]gmail[dot]com.