What Gertrude says. If the modem has only one "output" port, connected to the desktop, and you want to use both machines at once then you need a router. The Linksys one is good.
If you want to connect via wireless, then you also need a wireless access point (which will connect through the router, unless you get a router with built-in WiFi).
You should find that connecting the ethernet cable to the laptop, the existing software will do what's necessary.
There are some tests you can do: for example
-With the ethernet cable connected to the laptop, open a CMD box (click on "Start" and type CMD in the box, and press return).
- A black box should appear.
- In that box, type IPCONFIG /ALL
(note the space before the /)
You will get a screenful of information. Note the following bits and paste them in here:
IP address
Subnet mask
Default gateway
DHCP server
DNS server
From that information, you can tell a lot about what's happening.