Linking Arduino Nano with ESP8266 Wifi module
The easiest way to work with these devuces is to connect them to a spare pair of Arduino pins and use SoftwareSerial for communicating (unless you're using one of the Arduinos with more than one UART). SoftwareSerial work up to 9600 baud on 16MHz devices, 4800 on 8MHz devices, so you'll need to set the ESP8266 to accommodate that. They currently ship at 115200 baud. You can use the hardware UART on pins (0, 1) but then re-programming the Arduino means disconnecting &amp; reconnecting the ESP temporarily each time, and gets old pretty quickly.

Here's what I do with new ESPs (If you've re-flashed your ESP, you'll need to download the AT instruction-set emulator and re-flash it with that, first):

# To configure a new ESP module:
# Connect the ESP and the FTDI as in your top diagram, set your
# terminal-emulator to 115200 baud, and enter the following:

AT                             # ESP should respond "OK"
AT+RST                         # Resets the ESP
AT+UART_DEF=9600,8,1,0,0       # Sets ESP UART: 9600,N,8,1,no flow ctl

# Now change the terminal emulator baud rate to 9600

AT+CWMODE=1                    # Set ESP to station mode (client)
ATE0                           # Set ESP's echo off  
AT+CWLAP                       # List access points - after a short pause
                               # the ESP returns a list of WiFi access points


For future reference, here is a link to the ESP8266 AT Instruction Set.

Update June 2020: The above link was broken, now fixed. You can also find documentation at Espressif's github account (online readable, but not pdf, as of this date).