In simple terms, what you want is Broadband.
This will be either ADSL (Bt etc) or Cable (Virgin); which you go with will depend on who provides your home telephone.
Then you can choose an ISP - internet service provider - if you have a BT line, then you choose BT, Tiscali, Orange, etc. Again, who to choose is up to you.
Lets take it that you have a BT line at present a you are on dialup:
When you get your phone line activated, you need a modem / router. This will allow you to connect your PC to the internet with either a cable or wirelessly.
You can connect the main PC to the modem via a wire; laptops can be connected via either wires or wirelessly - ie so-called wi-fi.