MAC address and IP address are used to uniquely identify a computer. When is an IP address used and when is a MAC address used?
MAC (Media Access Control) addresses uniquely identify Ethernet network cards (or PCs with built in Ethernet). Although it's true that every computer that uses Ethernet has a unique MAC address, not all Internet connections are made over Ethernet. Also, MAC addresses in a particular local network are not similar in any way, so one can't use them to decide that a particular machine is "nearby."
IP addresses uniquely identify computers on the Internet, or on a local intranet. Computers on the same local "subnet," such as those on a particular local network, will share part of their IP address. For instance, computers in my house intranet are part of the 192.168.2.x subnet, with IP addresses like 192.168.2.6 and 192.168.2.22. The router that connects my house to the Internet has the IP address 192.168.2.1. That router also has a true Internet IP address, and performs NAT (Network Address Translation) to allow connections from computers inside my intranet to reach the outside world... but blocks incoming connections for security.
To communicate in an Ethernet network, MAC addresses are used. Ethernet frames (packets) don't know anything about IP addresses. ARP is used to find out the MAC address of a host on the local network. IP addresses belong to a higher OSI layer and are used to identify computers on one or more networks as the Internet is not one network, but a system of many networks.
An IP address is either given by a DHCP server (router) or you give it yourself. A MAC address is the hardware identifier for the NIC. This is static but you can alter it if you whish.
Your IP address is important since your network relies on it. Your mac isn't unless your ISP wants only one MAC address to connect with them over the modem. (totally offtopic though) Some people also use MAC lists as a security layer but this is highly insecure.