From LBPWiki

The term Acing or Aced refers to when you have completed a level from start to finish without dying or otherwise causing Sackboy to respawn from a checkpoint.

During a multiplayer game, a level is only registered as Aced if every player Aces the level together. It is for that reason that it is generally recommended to attempt Acing a level during single player games only, since Acing a level alone is considerably easier.

Benefits of Acing

Acing any level, whether in Story mode or online, will trigger the "Aced" icon to adorn its level patch from that point forward. In many instances, Acing a level will also unlock new goodies for your popit. These range from simple Stickers and Decorations to new costume pieces for Sackboy or even complex Objects to populate your levels with. However, it is only in Story mode that you can be be sure that Acing a level will unlock these items. When it comes to custom levels, it is up to the level's Creator to decide whether or not he or she will award the player with tangible items for Acing their level. Many times they do not do so, in which case the only reward is the "Aced" icon on the level patch and the satisfaction of having accomplished it. Also, because Aced means that you did not die, people who Ace levels often have the High Scores. Finally, Acing all of the levels (with the exception of those levels that only end when you die) earns you a Trophy in LittleBigPlanet and LittleBigPlanet 2, and is therefore a requirement for earning each game's Platinum Trophy.

Trivia

There's a little trick that will allow players to share the benefits on Acing a level without the need for every player to play through the entire level without dying. Simply have a single player Ace a level until the point just before the scoreboard, then invite any number of additional players into the game and proceed to the scoreboard to finish the level. Provided that the level has been Aced by the original player, any player that has joined the game should also receive the full benefits of Acing the level.

It is possible to ace Survival Challenges in LittleBigPlanet by turning off your controller just after starting the challenge. When the game asks you to "Press the PS Button to Play!", you will appear out of the Checkpoint, and land at the end of the level, Acing it. There is no benefit from Acing survival challenges, as they do not give you prizes for doing so, and are not required to earn the Play or Platinum trophies.

Note: This trick will not work in LittleBigPlanet 2

In LBP2 during the boss battles of each story mode "earth" if you ace the level then you will get the costume of the level's creator as well as many other prizes.