The reason for the split is that it's rare that you have a direct connection to the computer you want to send something to. The data you send usually goes to a router, which then forwards it to other routers, and so forth until a final router in the chain delivers it to your system.
For this to work, there must be a way of knowing which IP addresses can be reached just by dumping data out of the network card, and which IP addresses can only be reached "behind" a router. The network part of the IP address, determined by the network mask, makes this distinction.
So, if my IP is 10.1.1.42, and my netmask is 24 bits from the left (255.255.255.0), then that means I can reach any IP address from 10.1.1.1 to 10.1.1.254 just by sending data out of my network card. I'll use my IP as the source address, the end machine as the destination address, and just throw the traffic on the wire. It should get there if say, the other machine is connected via a switch or hub, without any further help.
Let's say I want to talk to 10.1.2.50. Well, I can't reach that just by spitting the bits out of my NIC. It has to go through a router. The simplest situation is where there is a default gateway given by DHCP. Let's say the default gateway given to us through DHCP is 10.1.1.254. I'll then send my traffic that I want to go to 10.1.2.50 to 10.1.1.254, putting my IP (10.1.1.42) as a source address and 10.1.2.50 as a destination address. The default gateway is not the ultimate destination, but I'm really expecting 10.1.1.254 to forward the traffic to 10.1.2.50 on my behalf. Hopefully the machine at 10.1.1.254 has a connection to the 10.1.2.0/24 network, and will be able to do that.
I hope this is helpful.