The thing is a small pot, as far as Arthur Dent can tell. It was given to him by his aunt, but he has no idea what it is or what its purpose might be. It says on the bottom that it was made in Ibiza.
It appears in the video game, where Arthur carries it around with him, despite losing it shortly before leaving Earth. It turns out that even when he throws it away or drops it, it returns to him. This proves useful during the course of the game, as things he puts in it will also return to him.