When someone has heart problems sometimes they feel pain in their left arm. But why is the left arm painful?
It has mainly to do with the embryonic origin of organs, with the heart being a typically left sided organ, it develops sharing some nerves with the left thorax and left arm.
There is however high variability, typically among patients but also among coronary vessels. For instance, right coronary stenosis may lead to abdominal pain, whereas left circumflex artery stenosis may lead to no pain at all.
Our pain-sensing neurons work in useful ways only when they inform about skin areas or muscles. When nettles sting your left hand, you want to retract your left hand right away. This is in part an educated reaction, because your brain ca re-learn what sensing neuron corresponds to what side of you hand when surgeons stitch together broken nerves.
With internal organs, it is not the same. You can't really retract your heart or your liver, and you can't look inside you to figure what organ is actually struggling. Our brains make sense of pain from internal organs by projecting them onto nearby skin or muscle. This is why doctors never write down "heart pain", but "pain projected to the area anterior to the heart" (in short, "precordial pain"). For them, it is never certain that the heart hurts. It could be even the pectoral muscles. (This is also why hypochondriac types are mocked when the complain about "stomach pain".)
Heart projects into many neighboring places. Some people complain about pain in the hand, but other describe pain in the throat or even the jaw. We feel something is not right, but we can't tell precisely where. The left hand is often the place where we project heart pain, because it is close to the actual trouble.
Pain from the internal organs is often, although not always, projected on the muscles and skin portions covered by the same nerve as the troubled internal organ. A more technical explanation describing which nerves cover the heart, and what skin / muscle structures these nerves are shared with, is given in Medical Physiology by Boron and Boulpaep.
The association between a heart attack and arm pain appears to be strange since the arm is usually not harmed. The reason for the pain in the arm is that the pericardium is innervated by the phrenic nerve and there may be some neuronal connections to left sided intercostobrachial nerves. During myocardial ischemia, the signals can overlap creating a false impression that the pain is coming from the left arm or jaw. The brain essentially gets confused about the source of the signal and this can often result in the pain from a heart attack being felt as pain in the arm with no sensation of pain in the chest. This is what is known as referred pain. You can check out this post on Wikipedia for more detailed information.