The brain is principally composed of about 10 billion neurons, each connected to about 10,000 other neurons.  Each of the yellow blobs in the picture above are neuronal cell bodies (soma), and the lines are the input and output channels (dendrites and axons) which connect them.
Each neuron receives electrochemical inputs from other neurons at the dendrites.  If the sum of these electrical inputs is sufficiently powerful to activate the neuron, it transmits an electrochemical signal along the axon, and passes this signal to the other neurons whose dendrites are attached at any of the axon terminals.   These attached neurons may then fire.

It is important to note that a neuron fires only if the total signal received at the cell body exceeds a certain level.  The neuron either fires or it doesn't, there aren't different grades of firing.

So, our entire brain is composed of these interconnected electro-chemical transmitting neurons.  From a very large number of extremely simple processing units (each performing a weighted sum of its inputs, and then firing a binary signal if the total input exceeds a certain level) the brain manages to perform extremely complex tasks.

This is the model on which artificial neural networks are based.  Thus far, artificial neural networks haven't even come close to modeling the complexity of the brain, but they have shown to be good at problems which are easy for a human but difficult for a traditional computer, such as image recognition and predictions based on past knowledge.