Introduce ${\mathbf Z}_p$ as "formal" infinite base $p$ expansions where you add and multiply by carrying (any other description will probably take too long and not be concrete). Show them the series for $-1$ in ${\mathbf Z}_3$ is $2 + 2\cdot 3 + 2 \cdot 3^2 + 2 \cdot 3^3 + \cdots$ by adding 1 to that, carrying, and killing off a new term at each step so the sum is 0. Then emphasize the idea that in ${\mathbf Z}_p$ the number $p$ is small and redo the previous computation with geometric series: $2/(1 - 3) = 2/(-2) = -1$. Show ${\mathbf Z}_3$ contains a square root of 7: $1 + 3 + 3^2 + 2 \cdot 3^4 + 2 \cdot 3^5 + \cdots$.

(To explain why $p$ being prime is important, say the $p$-adic integers form an integral domain, and for a contrast you could define $Z_{10}$ in a similar way and say there is a number $x$ in ${Z}_{10}$ besides 0 and 1 satisfying $x^2 = x$: $x = 5 + 2\cdot 10 + 6\cdot 10^2 + 9\cdot 10^4 + 8\cdot 10^5 + \cdots$. Compute the first few digits of $x^2$ to check this works. This is related to the elementary school question of finding integers whose square ends in themselves: $5^2$ ends in 5, $25^2$ ends in 25, and so on.)

You could introduce a topology on ${\mathbf Z}_p$ where numbers are close if a long string of initial digits are the same and make a metric from this too.  Then indicate how this makes ${\mathbf Z}_p$ compact by a sequential argument, in the same spirit in which $[0,1]$ is compact by an argument with decimal expansions. A key new feature here, which those with experience only in real and complex analysis haven't seen before, is that ${\mathbf Z}_p$ is a *compact ring*. In ordinary geometry there are plenty of compact groups, but no compact rings. 

I think a nice application of this compactness is the finiteness of integral solutions to certain equations. For example, $x^2 - 7y^2 = 1$ has an infinite number of integral solutions, but $x^3 - 7y^3 = 1$ has just two integral solutions: (1,0) and $(2,1)$. One way to prove this finiteness is to express the possible solutions using $p$-adic power series and use a $p$-adic analogue of a theorem your audience will know from complex analysis: an analytic function on a compact subset of ${\mathbf C}$ has finitely many roots in the set. Similarly, a $p$-adic power series that converges on ${\mathbf Z}_p$ has finitely many roots in ${\mathbf Z}_p$. Since ${\mathbf Z}$ is inside of ${\mathbf Z}_p$ this implies in particular that there are finitely many roots in ${\mathbf Z}$. To make this result effective (i.e., to know the integral solutions you found are the only ones), you one need  techniques to bound the number of $p$-adic roots of a $p$-adic power series, and that goes beyond the scope of your talk. :)