Connecting a desktop to the Internet without wires is a very easy process. It can be done in two ways.
1… A USB adapter / dongle can be bought that will connect a desktop to the Internet.
2… A dedicated wireless card can be inserted into the computers to enable Wi-Fi.
By far the easiest way is a simply buy a USB adapter that plugs in and after the drivers installed will allow wireless connectivity.
There are many wireless USB adapters on the market some as cheap $10 dollars and this is often the way most people connect a desktop wirelessly to the Internet.
The second option which is not as common since it requires opening a computer and some degree of technical ability. When this is done a computer card that is capable of Wi-Fi is inserted into the motherboards bus slot. The most common bus slot today is PCI. Running a dedicated internal Wi-Fi card has some advantages over an external USB adapter.
Besides the initial set up of having to open a computer and insert a new card, a internal wireless card does not have to compete for bandwidth as many USB devices do. If you have many devices connected by USB then you may experience slowdowns in internet speeds.
If you have a desktop that needs to connect to the Internet though wifi than the simplest way is to buy a USB adapter/dongle.
Wireless Desktop USB Adapters
Edimax EW-7811Un 150Mbps 11n Wi-Fi USB Adapter, Nano Size Lets You Plug it and Forget it, Ideal for Raspberry Pi / Pi2, Supports Windows, Mac OS, Linux (Black/Gold)
Panda Wireless PAU06 300Mbps N USB Adapter
TP-Link N150 Wireless Nano USB Adapter (TL-WN725N)