1) What is a neuron?
The brain is an organ made up of cells; just like any other tissue or organ in the body is made up of cells. The heart is composed of different kinds of specialized heart cells that are found nowhere else in the body, and so it is with the brain. The most famous types of brain cell are the neurons which differ from most other cells in the body in that they send out projections to meet each other to form a complicated network capable of fast and flexible communication. Other types of cell in the body communicate with one another in various ways, but only cells of the nervous system (including the brain) collaborate in large, fast, complex networks.
For a long time people thought that the cells of the brain were all joined up in a huge net; this was the reticular hypothesis (reticular means 'like a net'). But towards the end of the 19th century it became clear that the cells do not actually join up, and that each neuron was separate, a "fully autonomous physiological canton" (Cajal, 1888). Thus was born the neural hypothesis. (We will ignore, for the moment, all brain cells that are not neurons - although it turns out that this is a mistake.)
So, each neuron gathers information from its surroundings through thin thread-like input projections called dendrites. The slightly thicker (usually) projections that carry the results of the neuronal activity away, i.e. the output projections of the neuron, are called axons. Dendrites and axons are usually, but not always, on opposite sides of the main body of the cell which is the 'blob' in the middle.
The most obvious sort of activity that is carried away from the neuron by the axon is the action potential, which is a brief spike of electrochemical activity. These spikes, or clicks, are the usual form in which information is encoded and moved around the brain and travel down the axon (output projection) towards the dendrites (inputs) of other neurons.
In the places where an outgoing axon gets close to, but never quite touches, a receiving dendrite, there is a special structure that controls the passage of information across the gap, or cleft, called the synapse.
(My money, and all the smart money, is on the synapse, which is not yet that well understood, being the key to much of the really clever stuff that happens in the brain.)