To expand on William's comment:
${\bf Z}/n{\bf Z}$ is the integers modulo $n$. You probably know about modular arithmetic and if you don't you can find tons of information about it on the web and in texts about Number Theory and/or Discrete Mathematics. It doesn't contain the reals and it's not contained in the reals and the operations are not the operations in the reals.
Now pick a prime number $p$. Every non-zero rational number $a/b$ can be written as $(r/s)p^t$ where $r,s,t$ are integers and $p$ divides neither $r$ nor $s$. Define a sort of absolute value on the rationals by $|a/b|_p=p^{-t}$. This extends to a distance on the rationals by defining the distance $d(x,y)$ between $x$ and $y$ to be $|x-y|_p$. Now if you know how to get the reals from the rationals by putting in all the limits of convergent sequences, you can do the same thing but using $|\ |_p$ instead of the usual absolute value, and what you get is the $p$-adic numbers. Again, it's not a subset of the reals, nor does it contain the reals, and its distance structure is very different from that on the reals. Again, tons of info on the web and in (somewhat more advanced) Number Theory texts.