There are many people that can help you here. Typically, you have a line that comes to your house to a DSL or Cable modem which has an ethernet (LAN) out port which would go to your single computer. This will be on all the time and access to the Internet is instant.
People have become more computer literate over time and want more then one, up to hundreds that have the same access to the Internet.
This presents a little bit of a problem to the ISP (phone or cable) in that it requires more IP addresses which take bandwidth. Bandwidth cost the companies money, so they often will not alow you to have more then one IP address without you paying additionally for it.
People generally do not want to pay for any more then they have to, so Microsoft itself created a way to do "connection sharing" for internet access with one line. You can do this to set-up multiple computers, but the main computer that shares the connection must be on all the time for it to work. The sharing computer, basically acts as a software router for the other computers. If you wish to do it that way, look up connection sharing.
The preferred way to put multiple computers, thus a lan. Is to use a router right after the modem. You run a cable between the router (cat 5 or higher) and the modem connected to the line. Be sure you plug this cable into the right ports, it is pretty simple. Then you will need to run an ethernet cable from the router to each device (computer) you wish on your network. Run the software that comes with the router and you are set.
Routers are sold in various configurations, many only have one LAN port, this won't work so well unless you have a hub. So if you are doing a small LAN, you should get a router that has 4 or 5 ethernet ports, for 4 or 5 devices. You can always get a hub later if you need to expand more.
Routers have their ports labeled often with WAN and LAN. WAN or wide area network goes to the Internet side. LAN or local area network goes to your house or network side.
Hubs are basically trash, you should use a switch if you can afford it. Switching helps control network traffic, there is nothing worse then lag. Switches direct packets to correct location, whereas hubs broadcast to all ports, you can see that a switch is much more efficient.
Good luck with this and have fun!