Integrated means it's built onto the motherboard instead of requiring a (in this case) Network card to be fitted.
In regards to being wireless, that depends on exactly the description of the hardware. For wireless, it should say 'integrated wireless network card'. In your case the description indicates it being a standard network card e.g one requiring a wired network connection (ethernet).
Do you need a Router, this depends on if you intend to add other Computers to a home network.
I would always recommend a Router as they give you added protection to your home Computer(s). If you want to go down the line of a Router you'll need to give us some more information about your connection e.g does your ADSL or cable modem use a USB or a ethernet (standard network cable) connection to your Computer and do you intend to go wireless in the future.
