I guess my list would be:
Haydn
Mozart
Beethoven
Bartok
Shostakovich
Per Norgard
Schoenberg
Berg
Gubaidulina
Carter
And since this is a thread for the purpose of recommendation, I will make another list of 10, not necessarily my favorites (though some are very close), but ones that haven't been mentioned that I think are worth listening to.
Ahmed Saygun ( 3 & 1/2 Bartok-esque with Central Asian influences)
Witold Lutoslawski (Only one, but it's pretty epic if you like his aleatoric style)
Max Reger (6, these are good and meaty quartets but extremely dense with counterpoint and can get sort of academic, iono if you're into that sort of thing or not)
Gyorgy Kurtag (Multiple pieces for string quartet, but only one official "String Quartet" if ya know what I'm saying. Very similar to Webern in brevity and concentration, but with the Hungarian fire reminiscent of Bartok and Ligeti's 1st quartet)
Quincy Porter (A prolific American composer that hardly anyone cares about anymore. I have his 9 string quartets and, while they aren't the MOST brilliant things ever, I found them to be very enjoyable and chalk full of fun musical surprises)
Alfred Schnittke (Mostly known for his orchestral music and his other chamber pieces, especially the Piano Quintet. But he wrote 4 solid quartets that I very much like. Be warned though that, even if you are a fan of Schnittke, his first quartet is a very early work of his that was written back when he was experimenting with serialism)
Szymanowski (Wrote 2 awesome quartets. Very folk influenced and quite Bartokian. I'm referencing Bartok a lot, but he basically became one of the huge models for string quartet writing of the 20th century, especially for those composers interested in folk music as well)
Peter Maxwell-Davies (I believe he has written earlier quartets, but I'm referencing his recent series of 10 "Naxos" quartets. The only series of classical pieces I am aware of that is dedicated to a record label, these 10 quartets explore a strikingly broad range of expression and 21st century string writing techniques. They are quite complex sometimes, but they aren't as complex as Carter's quartets I would say. And they aren't "sound experiment" quartets, like those of Penderecki and Xenakis)
Gorecki (If you like the slow moving, long epic buildup style of his "Symphony of Sorrows" then his string quartets are pretty much that, in quartet form, especially the 3rd string quartet. There are some fast minimalism moments in these too, which can either be really effective or really mind numbing depending on what mood you are in)
Korngold (He is mostly known for his movie scores, his violin concerto and possibly his symphony too. But he also wrote 3 string quartets that are quite enjoyable. They reek of German expressionism, very similar to Berg or early Schoenberg, which is probably why I like them so much. The 2nd is known for its gracefulness and breezy Vienneseness)
EDIT: I just realized that someone had already mentioned Reger. So my list of composers that people haven't mentioned yet is only semi-accurate now, but meh.
In other news, I definitely second the Tanayev recommendation as well. Taneyev needs more love! His string quartets are great, especially once you hit #1. (Confused? Taneyev wrote SQ #7-9 before #1, and they are sort of meh, but his writing gets considerably better starting with #1 and ending with the brilliant #6).