Trying to be clearer...
OK - no you cannot re-use your modem I am afraid - it plugs into USB and is intended for single PC access to your DSL service. Thsi is what routers do - they allow the DSL to be routed to however many ports they accomodate. Apart from anything else your modem will plug into a USB socket - your router will only have NIC sockets on it (plus the 'phone line in).
You could put a NIC into the other machine, but then it would not be wireless - you'd simply be making a wired network. This will work with the router - but the point of routers is they have the DSL modem built in too, making the one you used redundant (I have two old ones myself).
Dongle is a commonly used term, but sorry if I confused you. It's not really the access point though - that will be a small box with the attached to the router that does the wireless bit for you - then the "dongle" is a dinky little device that just dangles (hence the name) from a wire into your USB port on any computer you're networking wirelessly.
I hope I've made things clearer.