Select About This Mac from the Apple menu, and then click the 'more info' button to launch the system profiler. Look down the left side and click on 'Airport Card'. If the pane on the right then reports 'no information found' you don't have built-in wifi, and would need to add wireless networking hardware to your mini in order to connect wirelessly to the router. This can be done by buying an internal Airport kit, which costs $99 and is hard to find, or by adding an external component such as the USB wifi adapter previously suggested, or by wiring the mini's ethernet port to a wireless 'bridge', which then connects to the router - or indeed, by wiring the mini via ethernet to the router.
If the pane lists details for the Airport Card, then you have one fitted, and should be able to connect to the router as long as the router is in range.
You will probably need to switch Airport on, which can be done in the Airport menu to the left of the clock at the top of the screen, if that menu is showing. It will appear as a fan shape, and likely is completely empty. If you see that, click on it and select Turn Airport On. The fan will then switch from empty to showing 4 light-grey arcs. You can then click the fan again and see a list of available networks. If you see the name of the router listed, simply select it, and if prompted for a password, enter that. Signal strength once connected will be shown by the number of light-grey bars in the fan then turn black - the more the better!
