Own wireless

Hello everyone.
I don't know much about internet/wireless setups, so I would really appreciate any help with my issue.

I recently moved to Japan where my internet provider (Yahoo BB ADSL through Softbank) makes me pay around $15(USD) extra a month for the privilege of having a wireless connection.

The router they set my connection up with is not wireless, so they basically have me rent a wireless card (I'm not sure if that is the term) which connects into the router from them.

I don't know what the card is called or if I can purchase one myself so I can save the $15 extra a month. Is there something else (maybe a USB thing) I can connect to the router to have a wireless signal?

If I buy my own wireless router could I replace theirs with it and still have my internet work?

Sorry for my ignorance, and thank you for any help.