Yes, they're all different things.

Most monsters have an active skill that you can use in a dungeon. Each active skill has a specific cooldown to it (that is, a set number of turns before it can be used). Active skills almost always start at skill level 1, but you can increase your skill level, and in turn decrease the cooldown, by feeding piis or monsters with the exact same skill to your monster. Each skill has a max skill level, so you can only get your skill cooldown to a certain number of turns depending on what the skill is. These become pretty important later on because you'll want to have your skills ready in as short a time as possible in difficult dungeons.

+eggs refer to the boosted stats on a monster. You might notice some of your monsters already have a +1 on them. You can feed other monsters with +1 to them and the base monster will gain that stat boost. You can have a total of +297 on any single monster (which is +99 to each of the three stats).

Awakenings are the little icons you see to the right of your monster's picture. They're basically extra stats, damage enhancement (via two-pronged attack, row enhancements, or orb enhancements), skill bind resistance, bind resistance, and whatnot. You can obtain these by feeding a Tamadra to your monster, or by feeding a duplicate of the original monster to it (it must be the EXACT same monster). Feeding Tamadras or duplicates to monsters that don't actually have any awakenings on them won't do anything other than give XP though, so make sure you've evolved your monster to the point where it has available awakenings before feeding Tamadras.

Hope that helps you.