OK, let's get right back to basics.

DCE = Data Circuit-Terminating Equipment; a modem

DTE = Data Terminal Equimpent; something that connects to a modem (eg, printer, computer, your MCU board)

The RS-232 spec (now officially known as EIA/TIA-232) deals specifically with the interface of a DTE connected to a DCE:

+-----+             +-----+
| DTE | <=========> | DCE | <----------> Telephone Line
+-----+    RS232    +-----+              (analogue)
         interface
         (digital)

The RS-232 spec defines a number of Interchange Circuits that form the interface between the DTE and the DCE; the two of interest here are:

Transmitted Data, TXD: This carries the data to be transmitted over the communications link (eg, phone line)

Received Data, RXD: This carries the data received from the communications link (eg, phone line)

+-----+       TXD        +-----+
|     |>---------------->|     |
| DTE |                  | DCE |<----> phone line
|     |<----------------<|     |
+-----+       RXD        +-----+

It should now be obvious that TXD is an output from the DTE, but an input to the DCE;
similarly, RXD is an output from the DCE, but an input to the DTE

Thus, when you use an RS232 interface in its intended application (Connecting a DCE to a DTE), you must connect the DCE's TXD to the DTE's TXD, and the DCE's RXD to the DTE's RXD. This is known as a "Straight-Through" connection.

However, when you connect two DTEs together, you must connect the TXD of one to the RXD of the other; this is known as a "Crossover Cable" - because of the way the TXD and RXD line "cross-over!"

+------------+                  +-----------+
|     TXD out|>--------+  +----<|TXD out    |
|            |         |  |     |           |
| DTE        |     +-- |--+     |       DTE |
|            |     |   |        |           |
|     RXD  in|<----+   +------->|RXD in     |
+------------+                  +-----------+

I trust you get the "crossover" idea there!


BTW: you may also hear of "Null-Modem" cables (and adaptors); these provide the TXD/RXD crossove, and also the necessary crossovers of the other control ("handshaking") lines.