Mathematical model on which current computers are built
The one closest to typical CPUs is probably the register machine or random access machine (RAM). A RAM has


an infinite number of registers, each of which stores an arbitrarily large number,
a set of operations on these registers (typically $\{+ 1, = 0\}$),
a programming language including these operations as well as control structures for looping/branching (until/if some register holds $0$) and
a program counter pointing to the next operation (in some program).


Real CPUs are quite similar, with some changes:


There are only finitely many registers (which may exist only virtually), and each stores only numbers of bounded size.
There are more operations.


Apart from that, it's very close indeed. It is common to extend the RAM model to account for memory hierarchy, which makes results a lot more applicable.