A bucket, also known as a pail, was a container with a handle meant for carrying liquids.
In 2286, Leonard McCoy asked Kirk if he was really going to attempt time travel in the HMS Bounty, which he classified as a rust bucket. Later, Scotty called the USS Excelsior a bucket of bolts. (Star Trek IV: The Voyage Home)
When Data experienced his first dream in 2369, he decided to paint pictures of it, but he ended up painting a bucket of water with smoke coming out of it. This confused him as he did not dream about the smoking bucket of water, but Geordi La Forge thought that perhaps it was because Data had dreamt about a blacksmith and blacksmiths sometimes used buckets of water to cool down pieces of metal. (TNG: "Birthright, Part I")
Before getting his own quarters, whenever Odo had to return to his liquid form, he would go into a bucket. When he didn't need the bucket anymore, he planted a plant in it in 2371. (DS9: "The Abandoned") However, in 2372, he smashed the bucket (and the plant) against the wall because he was angry. (DS9: "Crossfire")