let me try to explain...
There are 2 parts to an IP Address, Network portion and Node portion. Think of the network portion as your zip/city/street and the node as your name/house number. A computer uses a subnet mask to determine what parts of IT'S OWN ip are network/node. So for 255.255.0.0, the first two octets are network (the 255's) and the second two are node (the 0's). When you get outside of 0 or 255, you are getting into binary stuff, and it's a bit rough, I would suggest google for a decent site on bitwise anding
So, if you are 192.168.3.8, and the server is 192.168.1.7 (just making these up) then you need to use your SUBNET MASK to determine what part is network, and what part is node. lets assume for now, you have a subnet mask of 255.255.255.0. That would make your address 192.168.3 (network) and 8 (node). The server would be 192.168.1 (network) and 7 (node). So your PC looks at the network portion, and compares them. If they are the same, it sends a broadcast (who as X.X.X.X, tell X.X.X.X), if they are different, it goes to it's default gateway (please send this to X.X.X.X). In this example it's not the same, so it goes to the router.
Change the subnet mask to 255.255.0.0, and it's a different ballgame (or network). Now you have 192.168 (network) and 3.8 (node) looking for 192.168 (network) and 1.7 (node). The network portions match, so the computer assumes it is local, and just "hollers out the window" to find the local computer (aka broadcast, who has 192.168.1.7, tell 192.168.3.8).
Make sense?
Spidey is very smart, and confuses some of us n00bs from time to time. As you hang in networking more, you will realize what a good geek he is
