Also known as the "heart brain".
Your heart has ganglia, synapses, neurons, widely extended throughout its atria and ventricules, estimated at over 40,000 individual neurons in total (Armour, 1991). They are necessary to receive and transmit signals to and from the central nervous system and control basic cardiac function. The ICNS is a very active field of study, with major implications for a wide range of therapeutic contexts. The ICNS connects to the vagus (wandering) nerve, which carries a huge amount of information from the body back to the brain. There is a reasonable amount of evidence that the heart communicates to the brain more than the brain to the heart. Make of that what you will.
There appears to be a role for the ICNS in sympathetic nervous system activation in response to stress, pathological conditions and emotional states. I don't really want to get too far into that because I'm not well read/trained enough. If you're interested in this topic, definitely stick to reputable peer-reviewed journals, there is a lot of pseudoscience on this. It is wonderful to be able to imagine our own biological processes. No less wonderful is exploring the foundations for shared intuitions, like the idea that we think with our hearts to some degree (and our stomachs, and our skin, you get the idea). If you're interested in this stuff, have a look at Tem42's excellent intro to the enteric nervous system.
This write up had two purposes, to provide a very basic introduction to the existence of the ICNS, and to hopefully show just enough for the reader to be able to appreciate it beyond purely intellectual curiosity and understanding. When I first engaged with this topic, I was struck by the image of my heart, beating in the dark of my chest, working every minute of my life - making those minutes happen in fact, and silently thinking and feeling and dreaming. We each have one of these tiny machines, and they are beautiful beyond words.