Jam - extended musical improvisation over rhythmic grooves and chord patterns (the key word here is improvisation, which distinguishes these solos from those which are rehearsed).
Jam Band - an organized musical group with regular members whose live performances and albums are known for extended jams. The most recognized “Jam Band” in history is probably The Grateful Dead.
A Band that Jams - an organized musical group which records and performs structured, rehearsed songs but which often breaks into extended jams in a live performance. A good example might be The Allman Brothers Band or any number of jazz, blues or reggae bands.
The Monday Night Blues Jam - an event where musicians from many different bands (some of whom may have never met each other) spontaneously pick a song or groove and perform extended improvisational solos over it. This results in a unique performance each time, even if the song is well-known.