Improvisational comedy is where one or more performers present unscripted material, such as games, scenes, monologues, musical montages, what-have-you, before an audience; usually basing this material on suggestions received from the audience. This is generally done by a group of performers, although there are some instances of solo improv performers. In Europe, there is a tradition of "fooling," a solo form of "improv".
This is not sketch comedy (i.e. Mad TV, Saturday Night Live or The Carol Burnett Show). Nor is it stand-up comedy (Jerry Seinfeld, Richard Jeni, any comedian who stands at a microphone and tells a pre-determined set of jokes and gives substantially the same show to more than one audience). We respect and enjoy these forms, but they're not improv.
There are two major forms of improv that are typically performed:
short form and long form.
Short form improv is what Einstein Simplified performs. It is characterized by a series of brief (2-8 minute) improv games based on suggestions from the audience, driven by audience participation or given to the performers involved just before the game commences. A short form show usually runs 30 minutes up to 2 hours. There are pauses between each game as the rules are explained and ask-fors are retrieved.
Long form improv is more commonly seen in larger cities and was the initial format of an Einstein Simplified show. It is characterized by an initial suggestion, topic or idea garnered from the audience that drives the entire show. This is done by using either an idea generation segment at the beginning or by jumping in to a format and letting the ask-for drive the scenes that unfold. Not a clear explanation, but you need to see a long form show to understand better the concise definition given here. A long form show usually runs 1 to 3 hours, depending on the format chosen.
If you want to delve back into the ancient history of improvisation you must look to the Commedia dell'Arte that was performed in Renaissance Italy during the mid-16th century. This form was characterized by bawdy and risque slapstick during improvised scenes, with some characters wearing masks. Eventually stock characters appeared and the rest is history. How popular was it? Well it was enjoyed for about 200 years before it dwindled out of popular culture. It still did manage to influence Shakespeare and Moliere. There are hundreds of books and websites devoted to this form, so enjoy the research if you so desire.