Technically you just need a wireless AP (access point). A router allows multiple desktops/laptops to share a single Internet connection. A wireless router combines both functions. Normally ppl buy a wireless router since they typically need both capabilities. I assume at this point you’re NOT using a router (wireless or not) and therefore one of your laptops is connected directly to your cable/dsl modem (the other has no Internet connectivity at all). So a wireless router would certainly seem to be a good idea anyway.