Positive affirmations are simply: Statements which affirm something to be true.
This definition is still a little ambiguous so to elaborate, basically positive affirmations, what most people mean by them are positive phrases which you repeat to yourself which describe how you want to be.
The theory is that when you first start saying your positive affirmations, they may not be true, but with repetition they sink into your subconscious mind, you really start to believe them, and eventually they become your reality, they become a self fulfilling prophecy and actually become true.
Over time they overwrite any limiting or negative beliefs you may have about yourself or about not being able to do something, and replace them with positive thoughts and beliefs which instill confidence, belief, positivity, ambition and much more.