A ribbon tweeter is a metallic or metalised plastic membrane vibrating in a magnetic field. A conventional tweeter is pushing and pulling (or blowing and sucking!) just like a woofer, but is much smaller and hence faster.
Ribbons were delicate but are becoming tougher. Domes are more versatile and cheaper to produce.
I haven't heard a decent ribbon set-up at any length, though they have a dedicated, if small, following. Cannot express a meaningful preference, though having heard some amazing domes they are my default choice as there is such variety.