As well as the rather recent style of music mentioned in
Noung's writeup above,
garage is also the name of a style of rock music dating back to the 1960s and continuing to this day. (Incidentally, there is a node
garage rock, but most people who are fans of the genre just call it
garage, so that's the convention I'll follow).
Garage as a style was defined in the early 1970s by Lenny Kaye's epochal compilation Nuggets (full name Nuggets: Original Artyfacts Of The First Psychedelic Era 1965-1969). Kaye's compilation was something that had never been done with rock music before - a collection of singles that had never been hits and in many cases had only had an extremely limited release.
While Nuggets covered many genres, from the sunshine pop of My World Fell Down by Saggitarius, to the Beatles clone Lies by the Knickerbockers to the psychedelic blues of Baby Please Don't Go by The Amboy Dukes, the music that caught most people's attention was the thuggish, simplistic, but undeniably exciting music that made up the bulk of the disc. Tracks like Psychotic Reaction by the Count Five, Dirty Water by the Standells and Pushin Too Hard by The Seeds were labelled 'punk rock' by Kaye, but that designation soon became usurped by a burgeoning new genre influenced in large part by this music, and the term 'garage' became widely used instead.
A whole series of semi-legitimate and outright bootleg compilations followed in the wake of the legitimate Nuggets, with titles such as Pebbles and Rubble. From these compilations the genre became more and more formalised - a definition would have to include Vox amplifiers, distortion, cheap electric organ , and no more than four chords, as well as being recorded no earlier than 1963 and no later than 1970, and in America. This is pretty much the definition followed in the Nuggets box set (which collects many of the best tracks from these compilations, as well as the original album), and by modern admirers of the genre.
Garage is almost by definition a genre of tracks rather than artists - the whole genre is based around obscure one off singles by bands on a low budget - so naming 'important' or 'influential' artists is difficult, especially since most artists in the genre never heard each others' work - the bands were attempting (and failing) to imitate what they heard on the radio rather than consciously being part of a peer group as in other genres. A few bands - The Seeds, The Chocolate Watchband, The Monks, The Electric Prunes among them - have become retrospectively considered important, but even these produced few if any album-length works worth listening to. Some artists who were contemporary with the genre and could at a stretch be considered part of it would include Love, The Doors, The 13th Floor Elevators, the early Mothers Of Invention and the early work of Captain Beefheart.
Important tracks in the genre are easier to name. They would have to include the tracks listed above, along with 96 Tears by ? And The Mysterians , Louie Louie by The Kingsmen and She's About A Mover by the Sir Douglas Quintet among many others.
The influence of garage music continued long after the original bands stopped recording, simply because it requires practically no budget or technical skill to perform. Bands that were performing almost in the genre in the late 60s and early 70s, after the 'first wave' had died out but before the genre became widely recognised, include such later cult favourites as the MC5, The Stooges and the Velvet Underground.
The influence of garage music continued. In the 70s it was a huge influence on punk, especially the Patti Smith Group (which Kaye was a member of), Jonathan Richman and The Ramones, whose music was almost a caricature of garage music. The Cramps were particular conoisseurs of the genre. There also developed, in the 1980s, a rather bizarre movement of musicians who tried to copy the music note-for note, using only 'authentic' instruments. To my mind, this totally misses the point, but to each their own...
Over the last couple of years, the genre has undergone a revival, with bands such as The Strokes, The Hives, The Vines, The White Stripes and Stealth Munchkin (well, I couldn't resist... ignore the last one if you want) adopting the style - usually the bluesier end - to some commercial success (except Stealth Munchkin). These usually refrain from the more obvious 'retro' stylings of the 80s bands, and so are more interesting musically and will probably have a slightly longer artistic life.
While purists define garage music extremely tightly, most lovers of the genre use a far looser definition, and it overlaps in various points with psychedelia (especially) , blues, surf music, sunshine pop and folk-rock, so bands described as belonging to any of those genres may also be regarded as garage bands.
The first point of investigation for anyone new to the genre should be the Nuggets box and the more recent Nuggets II set. From there, you will soon be drawn into a twilight world hanging around backstreet record stores paying obscene amounts of money for obscure vinyl. Why this stuff is legal when crack isn't I don't know, but it's at least as addictive. But it's worth it for the occasional kick you get from a truly great track:
I feel depressed I feel so sad
'Cos you're the best woman that I ever had
I can't get no love I can't get affection
Oh little girl psychotic reaction
AND IT FEELS LIKE THIS...