What is the brightest star in a constellation called as?

Usually, but not always, it's called "alpha" + the Latin genitive form of the constellation's name.

If you don't know Latin, figuring out how to form the genitive is a little tricky, so you should probably look it up rather than trying to guess. For example, the genitive of "leo" is "leonis" but the genitive of "scorpio" is "scorpii."