Is is possible to set up a wireless network without a desktop as the server? My husband wants to buy a laptop to replace our old pc, but still have a home network.

The term "home network" implies you have more than one PC or networkable system such as Game console, Network printer, NAS storage device. The way forward for you depends on what equipment you already have and the type of internet connection you currently use. EG Do you have cable/DSL internet and what kind of modem? Is it a combination modem/router or do you have a separate router? If so is it a wireless router? If you have a wired only router you can either replace it with a wireless unit ( My recommendation would be one with both wireless and inbuilt 4 port wired switch), or a wireless accesspoint that plugs into a LAN port of your existing router. Now to your question on whether you need a wired server? No but it is generally necessary to use a wired connection to the router in order to initially set up your wireless network credentials and security protocol. You can switch to all wireless once this is done but if you need to access the router's admin interface in future this should always be done from a wired connection Goin' Fishin' (Some day)

"Is is possible to set up a wireless network without a desktop as the server?" Desktops don't normally act as a server for a network. I think you're a bit confused with the terminology. As Richard said, what type of internet connection are you using now and does it include the presence of a router (wired/wireless?) "So won’t you give this man his wings What a shame To have to beg you to see We’re not all the same What a shame" - Shinedown