RS232C separates equipment into Data Terminal Equipment (DTE) and Data Communication Equipment (Modems) (DCE). This is rather simplistic, as it always assumes that you will connect a modem to a terminal, however the use of the serial connection has extended somewhat since those days.
The RS232C standard defines the meanings of the signals, but not the type of connector, nor the pins on which each signal appears. Despite that, the 25 pin (and 9 pin) D connectors on an IBM compatible personal computer are generally accepted as a sort of standard for the pins, so I'll use these as examples.
When indicating a direction for data, it is usually from the viewpoint of DTE equipment. The popular names of the lines tend to be as follows:
D25 D9 Name PG 1 Protective Ground SG 7 Signal Ground TxD 2 (DTE) Data transmitted by DTE to DCE RxD 3 (DCE) Data received by DTE from DCE RTS 4 (DTE) "Request to Send" Start transmitter CTS 5 (DCE) "Clear to Send" Have started transmitter DSR 6 (DCE) "Data Set Ready" Modem ready to work DCD 8 (DCE) "Data Carrier Detect" Remote transmitter is active DTR 20 (DTE) "Data Terminal Ready" DTE indicates DCE may go off-hook RI 22 (DCE) "Ring Indicator" DCE says a remote DCE has called
The meanings of the signals relate to the days when phone systems were used manually, and when half duplex (signals in one direction only at a time) was common. Also, the connections went DTE (terminal) to DCE (modem) via phone line to (remote modem) DCE connected to remote DTE (terminal).
So for a phone call, the proper approach is activate DTR, send dial codes, wait for DSR to go active, and then look for valid data. To accept a call, wait for RI, only then activate DTR, and wait for DSR. If DTR has been active, the call is answered at once.
What about RST and CTS, and DCD for that matter, I hear you ask. They were only needed for half duplex operation. The DTE asserts RTS, and as soon as the DCE asserts CTS, the DTE can send. The remote DTE (as distinct from the local one) can use DCD to indicate it has valid data available on RxD (DTE should ignore the data on RxD until then).
Since half duplex became less common, it was noticed that CTS could be handy for controlling the data flow from DTE to DCE. However it would be nice to have DCE to DTE control as well, so some people used RTS as a sort of reverse CTS.
DTE asserts RTS, so DCE can send data DTE stops RTS, so DCE should stop sending data DCE asserts CTS, so DTE can send data to DCE DCE stops CTS, so DTE should stop sending but this was never how it was designed to act, because there is no hardware flow control designed in. These signals were for half duplex direction changes only.
You end up with a model something like this:
DTE DCE What it does TxD to TxD Data DTR to DTR Primary modem control RTS to RTS Secondary modem control Gnd -- Gnd Ground CTS from CTS Secondary modem control DCD from DCD Primary modem control RxD from RxD Data
When computers enter the picture, things change even more. You don't need DCD or any modem control between a computer and a terminal. But you do need something to indicate the other device is there.
See also this more comprehensive RS232 Standard explanation.