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I have installed Apache on my PC and am thinking of hosting a site from it.
I just wanted to know if I can get other LAN users to connect to my pc with a name (DNS?) instead of my IP address.
What exactly do I have to do to get them to be able to access it this way?

On Windows you can use the computer's name on the workgroup to access it. I'm not sure about Linux, there may be a DHCP server you can use or you can just update the hosts file on all the computers.

Do Unix networks have lookup names? That was an impression I got setting up Linux.

Using that name I am only able to use the name when I connect from my own pc.
From another guy's I had to type in the IP to connect.

If you're on a local LAN with Windows, both computers need to be on the same workgroup. If you're doing it over the internet you'll need to buy a domain name or you can use free services that redirect to your IP like Dynamic DNS or 2ya, etc.

AR wrote: If you're on a local LAN with Windows, both computers need to be on the same workgroup.

I am on a LAN.
BTW what is the difference between a Windows Workgroup and a Domain?

A workgroup is merely a collection of computers that are grouped under a name in order to categorise/organise them in a large LAN. A domain on the other hand features a domain controller server which is a central repository of user and computer policies, any user can log on any terminal on the domain (provided the Administrator hasn't locked them to specific terminals), user permissions are stored and given by the server. When the computer boots up it also downloads a computer policy for itself which defines what can and cannot be done locally.