Tenor trombones, usually, although there are bass trombones (longer, and lower range of notes...).
What do they play..? Any and all of the above, depending on the formation and musical style. In orchestras, usually they'll play the written score, which could be 'root-five' style or a complete harmony part. Listen to Schubert's 9th to hear how 'their own melodic lines' can be used. Glorious Stuff..! Of course, there's Dixieland, too, where, as long as a racket is being played (by everyone...), it's all good (or so I'm told...).