$\omega$ is defined to be the least number greater than all of the counting numbers, 0, 1, 2, 3, ... It has a number of names (the first limit ordinal, the smallest infinite cardinal, the first regular cardinal) reflecting its many pleasing properties, but of particular interest is that it allows even larger infinite numbers to be expressed: $\omega + 1$, $\omega.2$, $\omega^\omega$, etc.
The standard way to reason about these numbers (and their arithmetic) is set theoretic, and owes to John von Neumann. It is both very clever, and rather hard for computer scientists to reason about, so instead I will talk about ordinals exclusively in terms of nice sensible computer scientist-friendly _trees_, and show, amongst other things, why
$\omega + 1 > 1 + \omega$
and what induction looks like when extended to the transfinite.