Past few days ago I bought a Wi-Fi adapter for the Raspberry Pi, it just came today. When I tested it the Raspberry Pi wasn't recognizing the adapter, yet when I put it to my PC its perfectly fine. The adapter is a Realtek RTL8188EU. I'm presuming that I bought the wrong adapter.
Most Realtek Wi-Fi dongles do work on the pi, i had the same problem with a different Realtek dongle when i first tried to use it. You should start by getting an internet connection with a network cable and making sure it is up to date by using the commands :
sudo apt-get update
Once that has run you run:
sudo apt-get upgrade
After a reboot then you can check if the Realtek firmware is installed and if not it will install, by running the command:
sudo apt-get install firmware-realtek
One more reboot and hopefully your Wi-Fi will work.