An Ethernet network provides a way for a number of devices to communicate in a Local Area Network (LAN). This normally uses 10/100baseT RJ45 connectors and hubs or switches, but could include thick or thin Ethernet. The collection of devices sharing the same group of hubs and switches are all one Ethernet.
Packets sent on the Ethernet are basically either broadcast (every device gets it) or unicast (one device gets it). The later works by addressing the packet to a specific Media Access Control (MAC) address. All Ethernet devices have a manufacturer defined unique MAC address, and you should never encounter a duplicate. The MAC address is the computer's unique hardware number. On an Ethernet LAN, it's the same as the Ethernet address. In fact it is usually the address of the Network Interface Card (NIC) in the computer.
MAC addresses are 48 bit (6 bytes) and unlike IP addresses they are normally shown in hex. You may see them with colons between the bytes, e.g. 00:03:97:FF:00:09 but frequently they are shown as straight hex, e.g. 000397FF0009.
When sending packets to other devices on the same subnet it is assumed that the device is on the same ethernet. This is a key assumption in routing and ARP is used to find the MAC of the device with the require IP, and then packets are directed to that device.
When connected to the Internet from a computer (or host as the Internet protocol thinks of it), a correspondence table of the IP address to the computer's physical (MAC) address on the LAN is retained.
When sending packets to other devices, a gateway or router is needed. ARP is used to find the MAC address of this gateway and the packet sent to that MAC address.
See Ethernet Basics for a brief overview of the main points.