The simplest method would be to look at the back of your computer and see if you have a network connection port. The connection port will look similar to your modem phone line connection, but a little wider:

Modem Connection

Ethernet (network) connection and Ethernet cable

Another method would be to check the specifications that came with your computer. An Ethernet adapter might be descibed as:

10/100 Mbps Fast Ethernet Adapter 10/100 Network Interface card (NIC)
Network adapter
PCI Network Card
LAN adapter

Laptop computers may have:
Network connection
PCMCIA Ethernet card