To face Bowser, Mario must gather the Power Stars in order to harness their power (hence the name "Power" Star) to unlock doors that Bowser has sealed off in order to defeat him. There are 120 stars in the game, but only 70 are needed to defeat Bowser. In the DS remake, there are 150 Power Stars, but only 80 are needed to defeat Bowser.
The Power Stars returned once more in Super Mario Galaxy and Super Mario Galaxy 2, where there were once again 120 of them. However, rather than needing 70 to defeat Bowser, in Super Mario Galaxy you only needed 60. Once you collected all 120, you'd reap a much more satisfying reward (than Super Mario 64's) - the ability to play as Luigi. Collecting 120 stars in Super Mario Galaxy 2 would allow 120 more stars to be obtainable, all of which were green. Then after all the Green Stars are collected, the last galaxy would be unlocked (one star, one comet).