Israeli rock/difficult-to-categorize band (1973-1976; reunions: 1984, 1991, 1998, 2000*, and unconfirmed rumors say 2006 too but see update below!)
In no particular order:
While Kaveret may well be The Beatles on an Israeli scale, it's not "because they broke up and got back together multiple times", as corkscrewer said. Rather, it's because their music is Israeli music. Before Kaveret, no Israeli band had ever managed to play rock music without sounding ridiculously provincial, but they did not merely transcend that mediocrity: In the lyrics, the composition, and mostly the feel, Kaveret's music is at once world-class and undeniably Israeli, and in the best possible sense of the word at that.
They are perhaps best known for the incredible wit and humorous use of contemporary Hebrew in their every song, mostly the work of Danny Sanderson and Alon Oleartchik. Quotes from their hits (all of them around 30 years old, mind you, which is a lot for Israel's 56 years of existence) appear everywhere, and are recognized by everyone.
Kaveret represented Israel in the 1974 Eurovision Song Contest with the song Natati La Hayai ("I gave her my life"), finishing 7th. Considering that this was the year ABBA won with Waterloo, that's not half bad. The English version of the song was called "She Looked Me In The Eye", and was released in Europe with Morris And His Turtle - an English version of Hamagafayim Shel Baruch ("Baruch's Boots") - as the B-side. The English renditions are horrible compared to the originals.
This doesn't even begin to cover everything there is to say about Kaveret; "Inspite of Everything - Kaveret's Unofficial Archive" (available in both Hebrew and English) does a much better job: http://listen.to/kaveret
Update (May 04 2013 - whoa, 9 years):
Kaveret is reuniting! Or re-re-re-reuniting if you like. At any rate, 15 years after their last official reunion*, and 40 years since their debut, Kaveret is finally doing what fans have been bugging them about for, oh, around 15 years. Along with the series of concerts, Kaveret is set to release a 6 CD "anthology" of sorts, collecting alternate takes and other previously-unreleased material.
First, two consecutive small shows in Jerusalem were announced. Then they sold out, within hours of the announcement. Then another show in the same venue was announced. Then it sold out. Then a show in Yarkon Park (definitely not a small show) was announced, and promptly sold out as well. Then another show was announced, which, surprisingly, still has tickets available. This noder managed to grab tickets to the third Jerusalem show and is quite excited about this.
* The 2000 concert was hurriedly put together to raise money for band member Yitzhak Klapter's expensive brain surgery, and even the official press releases for the 2013 runion reunion seem to count from '98, saying "15 years" rather than "13 years".