The usual drum used in klezmer is a "poyk", which is just a small bass drum or tenor drum as used in a drumkit, played with the skins vertical and usually with a cowbell on top. The style is not particularly distinctive though you don't take off into soloistic riffs as a kit drummer in jazz/pop/rock might.

Sephardic players will probably use some form of Arabic frame drum but their lineups aren't as standardized as klezmer (and klezmer is anarchic enough).