If I understand correctly you have 2 devices connected via UART. I assume just TX, RX and GND lines connected between the devices? (i.e., no DTS/CTS/DTR/RTS control lines used - this is typical).
In this scenario, device 1's TX (transmit) is connected to device 2's RX (receive), and vice-versa. Their grounds are connected to each other. Thus, each device can be transmitting and receiving at the same time (each transmits on a separate wire, the communication is full-duplex).
The reason I mention all this is because it becomes clear that to "sniff" or "listen", you'll actually need 2 UARTs to listen to both sides of the conversation.
Basically, all you'd do is ensure that all 3 devices' UART GNDs are shorted, and connect (really, "tee", as in a T-fitting, like plumbing) device 1 & device 2's TX lines to the 2 RX lines on 2 UARTs. Make sure that the baud rates are all configured identically.
There are a lot of Arduino boards / designs. The most common one nowadays, the Duemilanove, uses the ATMega328P, which I think has only 1 UART (well, USART). So you'd either have to wire up a 2nd UART IC, or resort to "bit banging" on the second receiver.
Async UART communications is well-defined, with start & stop bits (and sometimes parity bits), so if your processor is fast enough, you can simply connect one of the device's UART TX lines to a GPIO configured as input, and poll the line fast enough with oversampling to detect START & STOP and sample bits. The article "Bit Banging" by Jack Ganssle will give you plenty to chew on.
A decent description of the RS232 waveform can be found at BeyondLogic.
Note that there are other issues such as voltage levels (0/+5, -10V/+10V, etc.) that you'll have to take into consideration (see the Beyond Logic section on "RS232 Level Converters"). I don't have enough information on your system to discuss hardware interfacing besides the "connect the lines" approach discussed above. Assuming voltage levels are matched, usually it's not a problem to "tee" the TX line into a second receiver (the sniffer), but if the TX doesn't have enough drive, you might need to insert a buffer/driver to prevent the signal from degrading.