Fought between Israel
from June 5th, 1967.
The three Arab countries neighbouring Israel - Egypt in the south-west, Syria in the north and north-east and Jordan in the east and south-east - were capable of joining forces for an attack on all borders that could deliver a death blow to the state of Israel (there's only sea on the fourth side).
Thanks to quick decision making and Pentagon-provided intelligence, the IDF managed to eliminate the threat in just six days, famously destroying almost all of the Egyptian air force on the ground in just two hours on the first day. The strike was a pre-emptive one - the armies of the Arab states were mobilising for action, but not yet fully deployed (hence the effectiveness of the surprise). It would be fair to say that the two superpowers (the US and the USSR) were playing at war over the heads of the people in the Middle east, as usual. The USSR fed the Arab nations only that intelligence which put Israel in the most dangerous and belligerent light, and likewise did the Americans with regard to their putative allies the Israelis. Both sides were of course arming their client-states.
The US ploy paid off better than the Russian one. Israel more than tripled its territory as a result of this war. The Sinai peninsula all the way down to the Suez Canal, the Golan Heights, the West Bank and the Jordanian half of Jerusalem were all taken with mind-boggling speed.
Immediately after the war there were already loud voices -- among them that of the often prophetic theologian, historian and philosopher Yish'ayahu Leybovich -- advising that the conquered land should be ceded back to Israel's neighbours in return for a disarmament agreement. Unfortunately, public opinion was vehemently against this, owing to a sense of overwhelming euphoria and invincibility the swift victory gave both politicians and citizens.
The Six Day War is a watershed in Israeli history. It provided a basis for much national pride, in a sense more than even the War of Independence. Unfortunately, the territories captured in the conflict and retained by the state of Israel are a bone of contention between it and its neighbours to this day.
It's easy to say that had the territories been given back to Syria, Jordan and Egypt immediately, much suffering would have been spared both from the Palestinians cought in the middle and from the Israeli nation itself. The Arabs' attempt to recapture what was lost in 1973 was disastrous and costly for all sides involved.
Hoewever, it is impossible to predict what the outcome of the Yom Kippur War in '73 would have been without the safety buffer of the Golan and the West Bank, not to mention the hundreds of kilometers of the Sinai desert. One must also consider the possibility that the psychological blow to the Arab nations is at least partly responsible for their inability to retrench and wipe Israel off the map shortly after the '67 conflict.
Those of us striving for peace can only hope that the political and military leaders on all sides have learned their lessons from the arrogance of the past.