Your thinking is a bit backwards: The iPod touch is portable, routers are not. Routers are devices that plug into the device that brings the Internet into your home (or hotel, business, restaurant, whatever) that turns it into wireless.
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0 w6 ]6 u" u% @The iPod is able to detect wireless access points whenever it's in range of one (the typical range of wireless networks is about 150 feet) and can connect to open (public) ones just by pressing Settings > Wi-Fi and selecting the network. A private network (identified by a "lock" icon next to its name) requires a password, which you can enter if you know it.
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Incidentally, there are a lot of apps available for the iPod touch that do not require network access in order to play. In fact, none of the games I play needs network access, and only a few of the apps do.