Trinket and Gemma are perfect for small projects needing to receive some external event, triggering your own defined output. This project uses the Adafruit IR Sensor to first receive IR commands from a remote, then to use those codes in controlling a project of your own.
To learn about IR signals and how they are decoded by a microcontroller, see the IR Sensor Tutorial from which this tutorial is based.
Many larger Arduino projects use the excellent IRRemote library by Ken Shirriff. This library allows for multiple protocols and is quite flexible. But it relies on hardware specific to larger Arduino processors. This project simplifies the process of obtaining codes and using them to scale to the limits of the ATTiny85 processor in the Trinket and Gemma boards.