A bucket, also known as a pail, was a container with a handle meant for carrying liquids.
According to James T. Kirk, Finnegan liked to play pranks such as putting a bucket of water on top of a door that was half open. (TOS: "Shore Leave")
In 2266, Ben Childress referred to Eve McHuron as being "plain as an old bucket". (TOS: "Mudd's Women")
In 2267, Montgomery Scott compared the effect of Vaal on the USS Enterprise's antimatter pods to "a pail of water on a fire". (TOS: "The Apple")
In 2268, Korax referred to the Enterprise as a "sagging old rust bucket" (TOS: "The Trouble with Tribbles")
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)
Despite Odo's inability to sleep, Quark suggested that Odo buy a latinum-plated bucket to sleep in. (DS9: "Q-Less")
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") Before leaving Deep Space 9 for the last time in 2375, Odo gave his bucket to Kira Nerys as a memento to remember him by. (DS9: "What You Leave Behind")