Can I use my laptop's wireless adapter in a desktop computer?

Assuming that the wireless card inside the laptop is a mini-PCIe card (which most are), it's pretty easy to fit it into a desktop. It's the exact same standard as PCI Express (just with a different connector and smaller form factor) and adapters are easily found on eBay:
http://www.ebay.com/sch/i.html?_nkw=wireless+mini+pcie+to+pcie+adapter
As you can see, most even come with an antenna or three already attached to the mounting bracket.

On the off-chance that it's an older laptop, converters from mini-PCI to PCI (sans the Express part) also exist, just modify that eBay search query accordingly.

For instructions on how to remove the wireless card, consult your laptop's service manual. The work needed to get to it may range from a simple removal of a bottom access panel to complete disassembly, but once you have access to the card, you just need to unscrew it from the motherboard and disconnect the antenna cables.

Usually the wireless card inside a laptop is a mini-PCI or mini-PCIe device. Desktops don't have slots that fit them.