After some studies for the last month, I have encountered a concept that has just completely eluded me, and that is Yield to Maturity (YTM).

Can someone explain the purpose and reasoning behind this concept?

My background is that I am going into a Finance Masters program after coming from a Math background.

I understand coupon rates, present value, maturity dates, and the general working of bonds and all that, but how does YTM work?

The general description my book uses is this, which barely helps:

The equation makes sense all right, but why would anyone need to figure out a discount factor (y in the equation) that makes the future cash flows of the bond equal to its price?

In addition, what is even meant by the "bond price"? As to this point in my studies only a bond's principal, face value, and par have ever been mentioned (and they're all the same thing).

I grasp that YTM is somehow supposed to account for other nominal yield calculations' deficiencies by incorporating Present Value, but it still confuses me.