(Actually, in the old days when "dumb" terminals were common, this terminology was even more confusing: both computers and terminals were "data terminal equipment"!)
Normally, two computers would talk to each other through modems and phone lines. The first computer sends some data on its "transmitted data" wire, its modem sends that over the phone line, and the modem at the other end hears that data and puts it on the second computer's "recieve data" wire. Note that the perspective shifted; the same data that was transmitted at one end became the data that was recieved at the other. And each modem arranged for incoming data to be on its "received" wire, and sent data from its "transmit" wire.
If you were to connect two computers directly together without a modem in between, you would have trouble. Why? Because they both want to send their data on the "transmitted data" wire, and will both listen to the "received data" wire. So their output circuits will often be fighting each other, one trying to force that wire to a logical "1" while the other tries to force it to a "0". Meanwhile, they will both listen on the "recieve data" wire, but neither of them will ever put any data there for the other to hear!
The solution, of course, is to cross-connect their transmit and recieve wires, so the transmitted-data wire from the one is connected to the recieved-data wire of the other, and visa versa. This arrangement is called a "null-modem", because it does the same job that a pair of modems would, without all the phone-line nonsense.
There is lots more to the RS-232 spec, but this is already most of what you need to know to get computers to talk to each other.