The Arduino Mini USB Adapter converts a USB connection into 5 volt TX and RX that you can connect straight to the Arduino Mini or other microcontrollers, allowing them to talk to the computer. It is based on the FT232RL chip from FTDI (drivers are included with the Arduino software).
Follow these indications if you want connect it with an Arduino Mini.
The circuit shown here is the basic setup for an Arduino mini connected to a USB-to-serial converter. You can see power and ground from the USB are run to the rails of the breadboard so it's convenient for the other components on the board. The 0.1uF capacitor from the reset pin is connected to the RTS pin on the mini USB adaptor. This enables auto-reset when the serial port is opened, meaning you don't have to press the reset button every time you upload new code. If it gives you problems, you can remove it, and press reset every time.