Newbie question on Wireless

Sorry for such a basic question, but I've been looking everywhere and I just can't seem to find an answer to my question (i.e., I don't think I'm understanding what I'm reading).

My questions is can a wireless router double as the following duties:

a) allow my computer to access the internet via wireless to another wireless router which is connected to the internet.

and -- not obviously at the sametime, but at a different time

b) allow another computer to connect to my router which is connected to the internet (and that router is connected to my computer via ethernet as well).

So basically, I'm asking if there is a device that will do double duty -- when I have it plugged into the internet, it will TX, and when I don't, it will RX. Example:

I'm at home -- I plug my DSL ethernet into the router. now my desktop and my roommate's laptop both can connect to the internet (via ethernet and wireless respectively). Now I run down to Panera Bread (or a hotel), I take my router with me, and plug it into my work's laptop via ethernet. I then search for the hotel's 802.11 connection, and away I go. Kludgy I know, but i'm curious.

Hope that helps. If a wireless router won't do it, is there a device that does that?

Thanks!!