Does Math.random() generate pseudorandom numbers uniformly?

In this comment in another question, dimo414 states that one of the problems with the OP's code is that it will generate random numbers non-uniformly. So I wonder why this is? Is it inherent the the particular algorithm? Or is it something about Math.random() itself? Is it because of the floating-point representation of the numbers chosen in the interval [0.0, 1.0)?

p.s. I understand the proposed answer to use Random.nextInt(). I also want to know more about the flaws of using Math.random().