No. That will connect your two computers to each other, but won't connect the laptop to the internet (or at least, not without some complicated messing about, and then you'd have to have the desktop on at all times you want to connect the laptop to the internet.)
Question: do you want the laptop to be able to access the Internet when the desktop is turned off? If so, you need a wireless modem and router and a WiFi card for the laptop (unless the laptop already has Wireless connection built in). That replaces your existing modem, whatever that may be.
There are some added advantages to that setup - the firewall in the router will protect you extremely well (as long as you set it up right!)
I've got such a network here, and both my daughters have similar in their houses. The "received wisdom" when I checked with the experts a couple of weeks ago is that the best wireless modem router for home use is the Netgear DG834PN - tech spec
here and Amazon's price
here.
If your laptop doesn't have a wireless connection, then I'd go for something like the Motorola WN825G (under £20 on Ebay) or the rather dearer Netgear WPN511 - see
here.
The desktop machine will need a Network card if it doesn't already have one - they are available for around a fiver, and fit into any spare slot in your PC.
If you browse around here, you'll find quite a few threads about setting up such a network, but it really is pretty straightforward if you work through the instructions.