For the computer itself you need some form of receiver. These come in two forms, either as a USB adapter you plug in (and install with a cd), or a card you have to install onto the motherboard of a desktop PC. Installing a card is best done by someone who knows what they are doing. The USB adapter can be used on a desktop PC or a laptop.
If you want your home to have a wireless signal you will need to buy a wireless router - there are tons of options and sometimes your internet service provider (Time Warner, Comcast, Etc) will give you a cable modem with a wireless router built-in! Otherwise, for public places with wireless all you need is an adapter to connect to the public wireless router.