I just did this myself - the router is much nicer than the modem and gives you something to "hide behind" so no more need for boring firewalls to administer.
You'll need the "always on" computer to have a NIC card - maybe it already has one? Do check... then a simple Wireless LAN access point that will connect to the boss computer and to the router. You can usually get these with USB dongles, that's the way I went.
So then you connect the router to the LAN and your computer to the Router usually (although you could leave that wireless if you so chose) and take your dongle to the other computer and you're ready to configure you network.
I went with EdiMax gear and have found it works fine for me - prices pretty good now too
PS You'll probably need a couple of straight through network cables with the situation I described.... Should be left with a spare dongle too if there's ever a lap-top around or anything.
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.