A crew is a group of people who usually live in the same area, and who all play Smash together. There is no set number of people required to comprise a crew, though three is generally a minimum since only two smashers would be considered a team rather than a crew. Some crews are more active than others, but there are basically three functions of a crew: Competing, Coaching, and Hosting. Many crews do all three of these while primarily focusing on one aspect more than the others.
Generally speaking, most of the Smash crews are originally formed solely for the purpose of competing in tournaments together as well as crew battles. These crews also travel all over to compete, often having to rely on each other for housing, food, and other expenses. Also, winning a tournament while in a crew generally makes the entire crew look good, effectively increasing the entire group's popularity in the Smash community, regardless of the gap between individual player's skills. Another reason playing competitively in a crew rather than by yourself is beneficial is that it's possible to make combo videos for your entire crew, which can be useful in creating multiple videos without a large amount of footage per individual players.
While not as common as competitive crews, some crews come together in order to increase the general skill of players in their community (or in the crew itself) as opposed to simply playing to win. While tournaments are generally used to help in this, many things such as written character guides and tutorial videos are created by these groups as well. This is considered the middle ground between the competitive side and hosting sides of a crew, because it normally requires a combination of both.
Crews are known to host their own tournaments, whether members-only or not, quite often. Few crews, however, are actually formed for that purpose (or have evolved into that purpose after a while). Many of these crews are responsible for the existence of the competitive Smash in their area, as they're the only tourney hosts in their general location. Sometimes, "host" crews simply revive a rather inactive Smash community, while at other times, they're behind starting up and maintaining the competition from scratch.