Ah, the book musical. The first thing that comes to most people's minds when they hear the word "Broadway", the requisite high school drama club activity, and possibly the most camp activity most people can think of.
Book musicals, which are primarily what people mean when they say "musical", are stage plays containing both music sung by the characters (usually original compositions, though not always) and dialogue spoken without musical accompaniment. The format grew somewhat organically during the 1800s, although pinpointing the moment where musicals became their own genre is difficult: due both to the temporary nature of stage productions and the fact that even separating musicals from other performance formats isn't foolproof, saying the late 1800s is the closest we're going to get. Wikipedia will tell you The Black Crook, but everything depends on where the lines are drawn: whether a show is an opera with some dialogue or a musical with both songs and spoken lines is a very subjective matter. The fact that operettas (which are typically lighthearted operas with notable dialogue, which is one good definition of a musical) and rock operas (which tend to just ignore all boundaries) exist doesn't exactly make it easier.
A good gauge of whether a show is a musical or not is if the spoken dialogue takes up roughly as much time as the songs do. This isn't exactly a perfect test - musicals are also sung very differently than operas, and tend to have shallower stories that are easy enough to follow without an outside guide. There are also the exceptions to this rule: one of the most famous stage musicals (and longest running), Les Miserables, is almost entirely sung through, and there are shows which have very minimal singing (the first drafts of musicals from people like Cole Porter tended to be roughly 1/3 sung through). The best test for whether something is a musical or not, then, is similar to the best test for things like people's religion or sexuality: the show is whatever the creators claim it is.
Musical theater tends to have a few unifying traits to it: the stories tend to be lighthearted musical comedies following a formulaic "guy meets, loses, gets girl" formula. The plot-lines are very often (theoretically, at least) slice-of-life in nature: perhaps due to the expense of turning human actors into anything else on a nightly basis, or the difficulty of having elaborate sets on a relatively small stage, most musicals are very human, day-in-the-life in nature. Musicals are also very frequently adaptations of other famous stories, due to the expensive nature of stage performances making most producers want to hedge their bets on an already successful property.
Of course, none of those rules are set in stone: especially in recent years the trend has been to distance yourself from musical tropes, resulting in (often depressing) musicals about (for example) the 1832 French Revolution or homicidal barbers. It's also worth noting that the most famous musicals tend to be very obscure adaptations of rather depressing stories; meaning that the common wisdom that musical theater is "happy or bust" might not be that accurate.
As always, the smaller the venue producing the show the more risks the show is likely to take. If you're in the mood for something off the beaten path, Off Broadway is the place for you.