Something that's beaten-up is worn out or falling apart from overuse, like a beaten-up old car with a loud muffler and rusty paint.
This word can describe things that have literally been beaten, or hit repeatedly: "The beaten-up boxer rested briefly before the next round began." It's also good for anything that's generally battered or dilapidated, like your favorite beaten-up sneakers or the ratty, beaten-up sofa that ends up in the basement TV room. You can also use the adjective beat-up as a synonym.