A Crate is an item that has appeared in the entire Super Smash Bros. series. Crates are containers which can hold various different items. When you pick up a Crate, because they are so heavy, you will not be able to jump or attack, and you will move very slowly while holding it. However, in the original Super Smash Bros., one could not move with it at all; this was a special ability available only to Donkey Kong. After the player has picked up a Crate, they can throw it up, forward, backward, or down. Crates are very similar to Barrels. If the player taps the control stick lightly, they cannot walk off ledges holding a Crate. But holding it down makes their character and the crate tumble off the ledge, and they drop the crate.
Exploding Crates are "dud" Crates that explode when destroyed and do not contain items. These Crates appear in all three games. One out of eight Crates are Exploding Crates. Exploding Crates deal around 49% damage and have strong knockback.
In Super Smash Bros. Brawl, Crates, along with Barrels, have a different appearance depending on what stage you are playing on. On a more futuristic stage, e.g Lylat Cruise, they take on a futuristic appearance, and on more fantastical stages, e.g Yoshi's Island, they look like wrapped-up presents. 
Item holders. You can destroy these to get at the items they contain, but be careful, because they will sometimes explode when broken. In Super Smash Bros. Melee, crates were uniform in design, but this time around, their appearance will change to match the stages. They'll sometimes look futuristic--other times, they'll look like presents with fancy ribbons.
Rolling Crates are a new type of Crate that appear in Brawl. These Crates are identified by their tiny, black wheels on the bottom, and can oftentimes go unnoticed. If a player hits this Crate, it will start sliding in the direction it was hit. Characters hit by the Rolling Crate will take damage and lots of knockback depending on the speed of the Crate, and the Crate normally breaks. Since characters can stand on top of them, players can take a ride on a Rolling Crate when it's been hit.
Special crates that look like normal ones but have four wheels attached to the undersides. However, what is more obvious is the "cart" that the wheels are attached to, and the fact that Rolling Crates will always be placed squarely on the stage, and never at an angle. Try to destroy these and they will go rolling along, smashing everyone in their path. When a rolling crate is destroyed, it will reveal the same sorts of items found in normal crates. You can also jump on a rolling crate for a fun ride.