You need hardware for two things: the connection to the outside world: the internet, and the connection between your computers. There are many choice for these tasks today.
For the connection to the external world the most common choices are:
an ADSL line, with an ADSL router or modem. It is the preferred way today: decent speed (almost), many commercial offers, choice between different contracts: you can pay basing on the connection time, the data you download or a fixed fee. Many network providers offer telephone service together with internet connection.
The old way: an analog modem connected to a telephone line; most provider still offer this type of connection; for a minimum fee or gratis. The maximum speed is 56 Kbit/sec, to low to look at the Flash commercial spots affecting to many web sites today, to download film, music, and Windows patch, but enough for downloading textual mails. Even updating your antivirus database can be an hassle with this connection. This is the best choice only when you have no other choice.
Radio waves (WiFi, wireless networks): in some zones there are public hot-spot offering internet connection with this technology. You have to be near to the hot spot to connect ( about 100 m ), the effective speed is around 10Mbit, depending on many factors, as the distance from the hot spot, the electromagnetic disturbance of the local environment, obstacles between you and the hot spot and the security measures used for the connection.
There are attempts to develop wireless technology to reach grater distances (many Km), as WiMAX; this could be the future, but is not the present.
Connecting to the unprotected wireless LAN of your neighborhood. This is a "pirate" choice and is deprecated.
For the connections among your computers (your LAN) you have essentially two choices:
put ethernet cards on your computers (if you are so unlucky to have a computer without a built-in ethernet connector), and use UTP ethernet cables for the connections between your computers. This is the safer and most performant choice, but you have to deal with cables all around. A speed of 10, 100 or even 1000 Mbit/sec can be reached, more than the speed a single computer can use. This is a mandatory choice if you must reach a computer in a far or shielded zone of your house. The maximum distance of this connection is 100 m; to go further you need some kind of signal repeater.
Use wireless cards, and use radio waves. Less performant, prone to security risk, but preferable if you don't like cables at all or use mobile computers. The maximun distance reachable depends on many environmental factors, usually you con safely reach distances of 20 meters or more with a speed of some Mbit/sec. More far and more distance means less speed. New technical solutions (named 802.11n protocol network) promits very high speed, we will see, maybe next year or so.
With a wireless network you can easily cover your apartment, but also parts of the surroundings, you have to crypt your connection to avoid incidental intromission into your net, and this halves your speed.