Broken chords (also referred to as arpeggios) are three or more notes that aren't played at the same time but closely enough to still be heard as a group or whole. And even the three-note rule is open to the occasional exception; some guitar chords consist of only two notes, but they still function as chords because they work diatonically in the same way that a major or minor chord would.
Guitar chords might very well be the most important element of guitar playing; after all, they're the basis of what makes a song. Most people picking up a guitar for the first time figure out a few guitar chords before even going for their first lesson, and still more teach themselves guitar without any help from an instructor. Self-taught guitarists learn chords in a number of ways. Some learn by listening to their favorite songs and slowly picking out the notes, a common yet often frustrating process. Others figure out the chords by learning to read tabs, a type of sheet music intended for fretted instruments that uses a graph-like chart to show where on the frets the fingers are placed. Both techniques are common among those learning the guitar, though the number of self-taught guitarists who never learned to read tab is fairly high.
Just like any other instrument, the sheer number of possible chord-groups can often be overwhelming for a new guitarist. And even the frequently taught chords are beginning to fall by the wayside, making room for a variety of guitar sounds created by tuning the strings in almost innumerous ways. Though power-chords (a type of note group using a base note, an octave note and the fifth) are still the most common type of chords, new bands are increasingly experimenting with alternate tunings to create new sounds; alternative bands have been toying with this way of playing interesting guitar sounds for decades.
So how many chords does a guitarist really need to know?
Most simple songs contain just 3 called "primary chords". So even a stark beginner can learn 3 simple note groups well enough to strum along and accompany himself as he sings. But after that, the sky is the limit; there are thousands of possible chords, so it is up to the individual guitarist as to how many he or she want to master.