Can I use a serial module with a barebones Arduino?
The UART is the piece of hardware the transmits and receives data serially.  It takes a byte of data (in parallel form) and then shifts it out 1 bit at a time.  For receiving, the opposite occurs:  after 8 bits of data come in, a byte is made available.

There is a UART inside of the ATmega328, which is the microcontroller on the Arduino.

When people say "connect to the UART of the Arduino" they are referring to pins 0 and 1.  [Refers to Uno, Leonardo's Serial1.  Mega has 4 hardware UARTS]

The Bluetooth module is probably meant to be connected to pins 0 and 1.  Then when you do a Serial.print("Hello") or Serial.write(0x55) in the Arduino code, that data will be transmitted over Bluetooth.  

Bluetooth devices such as this one, generally use the SPP (Serial Protocol Profile) to appear as virtual serial ports on the host device [PC].