Paraphrasing from Mixerman:
"A ribbon has a diaphragm that's made out of ribbon instead of a capsule that's made out of whatever it is that capsules are made out of."
Ribbon designs have changed a lot over the past few years so there are a lot of generalizations that don't apply to a few of the newer mics.
Ribbons are typically sensitive to air movement and can be damaged easily this way. Same thing to an extent with certain condensers. You wouldn't typically want to put one in front of a sound source that could blow out a match.
Do not blow into your ribbon mic.
They can be damaged by the presence of phantom power under a few circumstances - be careful if you're hot-plugging mics into a TRS patch bay with phantom running through it. Better yet, don't hot plug anything into a TRS patch bay with phantom running through it.
Sonically they can be voiced all over the place. Bright. Dark. Whatever that means.
Ribbons tend to be a bit mellower sounding than some condensers or moving coil mics. They take to EQ really well. Good ones through a high quality preamp usually deliver a very smooth sound quality.
Ribbons are well known for capturing a really wide range of frequencies without being overly hyped in the high end. Some condensers are kind of touchy that way. Transient response is typically faster than moving coils and slower than condensers. Detail is very good. Most of them are figure 8 devices, so rejection at the nulls is excellent and proximity effect is usually on the large side.
People use ribbons a lot for things where they want good high end detail without screeching harshness. They're usually very good on drum overheads and wind instruments but can sound really good on other sources as well.