What is IRI

The IOTA Reference implementation, commonly known as IRI, is the reference implimentation of the IOTA 'node' software in Java.
The IRI software enables the Tangle to operate by enabling individuals to run the software and operate a Node.

The Node

Running the IRI software enables your device to communicate with neighbours in the peer-to-peer network that the Tangle operates on. Any device connected to the network.
The official IRI Github repository can be found here: https://github.com/iotaledger/iri

Why run a Node?

It encourages decentralization of the network, and removes the need to trust a third party node to access the Tangle.