What are the dots inside the eyes when someone uses Sharingan?
These dots are called tomoe.  


  When first awakened, each Sharingan usually will have only one tomoe (巴). Through training and continued use, the Sharingan will develop a second tomoe and then, upon full maturation, a third. All the Sharingan's abilities are available to the user from its earliest stage, but with greater development comes greater proficiency with those abilities.


For example during their first fight in the Valley of the End Sasuke was able to easily see Narutos movements after obtaining the third tomoe in one of his eyes, when he couldn't dodge them at all earlier.