Here are some common classifications, but keep in mind that there's overlap and many categories:
By Temperature and Color:
* O: Hottest, blue
* B: Very hot, blue-white
* A: Hot, white
* F: Moderately hot, yellow-white
* G: Our Sun's type, yellow
* K: Cooler, orange
* M: Coolest, red
By Size and Mass:
* Supergiants: Largest and most massive
* Giants: Larger and more massive than the Sun
* Main sequence: Average size and mass, like our Sun
* White dwarfs: Very small and dense remnants of stars
By Evolutionary Stage:
* Protostars: Stars still forming
* Main sequence: Stars fusing hydrogen into helium
* Red giants: Stars that have exhausted hydrogen and are fusing helium
* White dwarfs: Dead stars that have stopped fusing
* Neutron stars: Very dense remnants of supernova explosions
* Black holes: Extremely dense objects with such strong gravity that nothing, not even light, can escape
By Properties:
* Variable stars: Stars that change brightness
* Binary stars: Stars orbiting each other
* Neutron stars: Stars with extremely high density
* Pulsars: Rotating neutron stars that emit radio waves
So, depending on what criteria you're using to classify, the number of "main types" of stars could be as few as a few or as many as dozens.
