The name is your computer's hostname, which is sometimes, but not necessarily always the same as the name you gave it when you set it up for the first time. It does sound as though you have been given another hostname by DHCP. The hostname is one of the optional settings that a DHCP server can assign, but does not have to.
DHCP is a protocol that is used for a computer to ask a DHCP server on the same network to tell it all the settings needed to connect to that network, instead of entering them manually. Without it, you would be unable to connect to a network without knowing how to configure your computer with an IP address, subnet mask, default gateway, DNS servers and other settings that would work with your particular network. If you don't understand what those terms mean, that's exactly the point I'm making.
I've had experience of the DCHP server running on Windows 2K server assigning to macs the hostname of the previous Windows XP workstation that had been using the same IP address. (The sysadmins told me that was because the mac DHCP client was broken. I have my doubts.)