Odo's bucket was an object that Odo used to store himself when he had to revert to his natural, gelatinous state to regenerate. He generally stored it in the back corner of his security office but would take it with him on trips off the station expected to take longer than 16 hours (the length of time he could remain in solid form).
Odo took the bucket with him on a trip aboard the USS Defiant to the Gamma Quadrant to meet the mysterious Dominion. He had to share quarters with Quark and made the Ferengi look away when he melted into the bucket. (DS9: "The Search, Part I")
After his encounter with the Founders, Odo stopped using the bucket; he would instead revert to his gelatinous state and shapeshift freely throughout his quarters, allowing himself to spread across the floor when he needed to become gelatinous. Odo kept the bucket with him "as a reminder of how [he] used to be." He put a houseplant that Major Kira Nerys brought him as a housewarming gift and decoration into the bucket. (DS9: "The Abandoned")
When he believed he was about to die on a cold mountainous planet in 2373, recorded a log in which he stated his wish to be cremated (having been transformed into a Human at the time), and his ashes placed in his bucket, which he then asked to be shot through the wormhole. (DS9: "The Ascent")