TCP/IP v4 network address used for broadcasting a packet to every available subnet. The use of 0.0.0.0 has been deprecated by the IETF in favour of 255.255.255.255. This won't work over the Internet, because if your Internet provider is worth its salt then it will drop any such packets (and if they don't, others will).
In the context of a network routing table, 0.0.0.0 means "default route", sometimes refered to as "the route of last resort". When a program sends a packet to an address that does not appear on the computer's routing table, the packet is forwarded to the gateway for 0.0.0.0, which is hopefully able to route it to the correct address.
My computer, XERCES, is connected to my family's main computer, SHODAN. SHODAN shares our cable modem connection with NAT (implemented as a part of MS Internet Connection Sharing).
XERCES' IP address is 192.168.0.2, and SHODAN's is 192.168.0.1. When I run route print on XERCES, this happens:
0x1 ........................... MS TCP Loopback interface
0x1003 ...xx xx xx xx xx xx ...... Realtek RTL8139/810X Family PCI Fast Eth
Network Destination Netmask Gateway Interface Metric
0.0.0.0 0.0.0.0 192.168.0.1 192.168.0.2 1
127.0.0.0 255.0.0.0 127.0.0.1 127.0.0.1 1
192.168.0.0 255.255.255.0 192.168.0.2 192.168.0.2 1
192.168.0.2 255.255.255.255 127.0.0.1 127.0.0.1 1
192.168.0.255 255.255.255.255 192.168.0.2 192.168.0.2 1
22.214.171.124 126.96.36.199 192.168.0.2 192.168.0.2 1
255.255.255.255 255.255.255.255 192.168.0.2 192.168.0.2 1
Default Gateway: 192.168.0.1
Notice that any packets destined for 0.0.0.0 are passed to 192.168.0.1 (SHODAN). If I run a program that, say, wants to talk to www.apple.com, it will send packets to 188.8.131.52. Since this address does not appear on the routing table, the packets are sent to 192.168.0.1 (SHODAN) instead. SHODAN will see that 184.108.40.206 isn't on its routing table either, and pass the packet onto its gateway for 0.0.0.0, which is 220.127.116.11 - the Universal Broadband Router that the cable modem is connected to.
Finally, in some cases 0.0.0.0 can be used to refer to "this computer". For example, requests for DHCP information are made up of regular TCP packets and as such have a field in the header for the source IP address. However, the chances are that a computer sending a DHCP request will not yet have an IP address: in this case the address 0.0.0.0 will be used.