MAC addresses are part of the link layer of an Ethernet network. The same link layer segment of a network must have unique MAC addresses so that packets can be unambiguously routed to the correct destination. MAC addresses actually consist of two parts, one part is assigned to the manufacturer, the other part is basically a serial number that the manufacturer generates. So from looking at a MAC address, you can look up who manufactured the device. The MAC addresses are not broadcast beyond the layer 2 network segment that you're connected to. However, that does mean that anyone in range can read the MAC address of your wifi card as it is sent 'in the clear' (not encrypted). Generally MAC addresses are also used for access control (though this is not particularly effective as MAC addresses can be spoofed) and for long-term IP address assignments (same IP can be assigned on subsequent connections).
Yes, computers broadcast their MAC address when they first connect to a network so that they can, for example, obtain an IP address via DHCP. However the MAC address isn't forwarded beyond the LAN to which the computer is connected.