For my home network I took a basic desktop PC and added 2 large drives on RAID1. Then I shared the drive over my network for file sharing/backup purposes between my computers. I also hooked up my printer to it, that way I can print from any computer, and since my server is always on I don't have to bother to turn another computer on to print. That's the basic thing you would use a home server for. Many people do more advanced things though.
Never heard of laptops as a server and it wouldn't be a good idea. For one, you are limited to 1 drive. And generally servers stay up 24/7 for ease of access. I wouldn't want to leave a laptop on 24/7. An old P3 desktop running Windows 2000 Pro is quite enough to run a file server for your home.