Speeds above 300 are only needed if you are processing pictures or other forms of graphics. So buying a new computer is not going to do much for your slow Internet speeds unless you are still using a slow modem. The box that contains the CPU is frequently called the CPU because it contains all the components that do the processing. The boxes are available in desktop and tower format. Desktops usually sit under the monitor.
Towers usually end up on the floor.
This is the large circuit board that everything else plugs into. It carries the electrical information between the various parts as the information is processed. The various parts that may be included or plugged into it are the Modem, video card, network card, and places to plug in the mouse, keyboard, scanners, cameras, etc. These plugs are called Ports and are usually on the back of the CPU box, although many computers have some plugs in front. These are small circuit boards with some parts called chips.