HC-05 is  a cheap (and easy to use) module that can be used to provide bluetooth connectivity to Arduino.

There are cheap breakout boards available on ebay that can be powered directly by the 5v supply of the Arduino. Below is a picture of such a breakout board. Though my board has “level 3.3v” written but it is working fine on 5v I/O levels. I haven’t run it continuously for days yet but at least a few hours didn’t kill the chip.

Some of these boards has one problem - the EN/Key pin, of the board, required for programming the chip (through AT commands) is not connected to the EN/Key pin of the chip. To program the chip on such a board, manual shorting is needed (using some wire) between the EN/Key chip on the board and the chip. In the above picture the EN/Key pin has been marked.

The circuit is pretty simple, connect the VCC to the 5v supply of Arduino, GND to GND, Rx of the module to Tx of Arduino and Tx of the module to Rx of Arduino. For the Rx and Tx software serial ports can be used also.

The same circuit goes for HM-10 too, which is a BLE device.