Side-Levels are optional levels that are unlocked by playing through the main story, usually through Keys, though LittleBigPlanet 3 introduced Side-Quests, which are unlocked by finding quest-givers in the hub levels. They tend to be shorter and simpler than the main levels. In LittleBigPlanet, Side-Levels did not unlock any Prize Bubbles. Apart from fun, the only incentive to play them was to unlock the "Play" Trophy. LittleBigPlanet 2, however, introduced the Score Sensor, and since then they have regularly offered prizes for getting a certain score.
Score Challenges are usually structured like a mini story level. In them, you try to get to the end of a level while getting as big a score as you can. This could involve anything from Races to jumping through hoops. In some, e.g. Tower of Whoop and No Drain No Gain, it's possible to get every bubble in the level in one long score multiplier.
In Survival Challenges, the goal is to stay alive for as long as you can where you try to stay alive as long as you can while collecting points. They only end when you die, and are respawned near the Scoreboard.
LittleBigPlanet 2 introduced Versus Levels, which are designed for two or more players to go against each other, and whoever gets the highest score wins. Unlike the other types, they do not offer any prizes.
Contraption Challenges were introduced in LittleBigPlanet 3. In them, the player uses the Popit Power-up to build a vehicle, and then puts it to the test, e.g. racing Gustavo in The Wheel Deal or "Air Jousting" in Joust in Time.