The main answer is through the blood (circulatory system) and through various different types of neurons and nerves (nervous system). Human blood isn't just one chemical, it's actually hundreds of chemicals all being pumped along different channels. Oxygen, Carbon Dioxide, Blood cells, Hormones, Glucose and more are all carried from one cell to another in the blood, sort of like a highway in your body down which many types of vehicle can pass.
But a lot of the signals between different parts of the body are done via electrical signals sent along nerves and neurons. Your skin (arguably the largest organ in the body) is wired into your brain via the nervous system and can send it information about pleasure, pain, temperature and texture, all of which your brain processes, sending its own signals either down the nervous system or by triggering various glands (little chemical factories) which pump chemicals into your blood telling your body how to respond.
And these are just two two main systems. Your organs can also "talk to each other" indirectly by changing different parts of the body around them. Your muscles can produce acid which gets carried to the liver (mainly by the blood) and the liver responds by digesting it into less damaging chemicals, almost as if the organs have sent a message to the liver telling it to process the acid. Your body is really a network of different systems all communicating with each other whether you're aware of it or not. They do it in your sleep, when you're awake, when you're thinking about it and sometimes by accident.