Question

What determines which specific atoms of a radioactive material decay over a period of time?

Answer 1

The half-life is the amount of time in which specific atoms of a radioactive element will go through decay. This is used as a measurement to estimate how long will it take for one-half of the atoms in a sample to undergo radioactive decay.
Moreover, all radioactive atoms undergo radioactive decay and will continue to transform into new elements until it will reach a non-radioactive and stable isotopes.

Answer 2

The answer is: "nothing".
In fact, it's totally random. When there is a certain amount of atoms of a radioactive element, it's not possible to predict which atom will decay first and which one will decay later. We can only predict the amount of atoms that will be left after a certain time t (if we know the half-life of the element), but we can't predict which atom will decay first.