If your laptop includes a wireless NIC, you’ll be able to connect to the Internet anywhere that you can get a wireless (WiFi, 802.11b/g, etc.) signal. The availability of a signal depends on your proximity to a transmitter, which — yes — is usually a router. Many coffee shops, airports, and other public venues have WiFi “hotspots” set up for your use (which literally means that they’ve got an industrial-strength router plugged in behind the counter or in the ceiling somewhere).

At home, you’ll need to buy your own router unless

1. your cable/DSL modem includes a WiFi transmitter (which is rare), OR
2. you’re close to neighbors who have an unsecured wireless network

Unfortunately, using your neighbor’s network is often illegal (or at least against their ISP’s ToS), and unless you’re in an apartment building, the signal is unlikely to be reliable enough to use.

Let me know if you have other questions.