Just to add to what Morsing said:
The reason we call it a "subnet" is because that's what it is. A "sub-network", a network within a network ID.
In the case above, your network ID is 192.168.1. However, when you "subnet" you are breaking this network up into pieces. Think of a pie. The pie represents the network ID. If you cut it into equal sized pieces, each piece is a "subpie". The more "subpies" you make, the smaller each "subpie" becomes since they all must be the same size (if they weren't, everybody would fight for the bigger pieces and whoever is holding the knife would be tempted to stab everybody...)
Putting a computer at 192.168.1.20 keeps the computer in the same Network ID but it exists in a different subnet. This computer cannot "talk" to the other computers directly. It is sitting on a different piece of pie. In order to communicate with the others, it needs something to act as a connection point (router or gateway). Without this, it will just sit there and grow old, eventually dying of loneliness.
Now, in the real world, we don't bake pies when connecting computers (but wouldn't that be a better way? I digress...). Instead we run cables from computers to things like hubs and switches, etc. So, let's assume we have a 12-port hub. We run a cable from all the computers to the hub. The computer at 192.168.2.20 is connected to the same hub as all the other computers. Looking at this setup, we ask ourselves, why can't this computer communicate with the others? After all, IT'S RIGHT THERE! They all share the SAME HUB!
Essentially, computers have 2 methods of communicating with other computers. Directly (by sending a message to a specific MAC address) or "Fundamentalistic Evangelism" (screaming at the top of their lungs to everyone in earshot, also called broadcasting in network-speak).
The problem is that one of the reasons for the existance of IP is to organize addressing of computers. This is a "logical" organization as opposed to a "physical" one. Even though this rogue computer is physically connected to the others, the logics of IP mean that this computer may as well be on the other side of the world. It's invisible to the others.
(However, because a physical connection exists, you can "cheat" the logical IP addressing by simply going to this computer and finding its MAC address. You can then covertly communicate with this computer by making packets that contain that computer's MAC address. I've never tried it but you MIGHT be able to add this computer's address to the ARP table manually which would allow communication. If you don't know what I'm talking about, that's good, you won't be shot.)
Finally, why would you want to have separate subnets anyway? A bigger question may also be, why do I have so much time to type all this drivel? (I'm afraid to answer that one)
Breaking up your network into different subnets can serve many purposes but probably the biggest is to logically separate each department. The finance department would be one subnet, the engineering department another. A third subnet could be for the pimply-faced nerds in the company to play online games among each other without those stupid "executive-types" knowing what is going on.
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