Neurons are the cells that make up your nervous system, including your brain and spinal cord. A schematic is shown above, as well as a photograph of some actual neurons. One of the most important things about neurons is that they form a network, via a large number of connections they make to other neurons.
There are several different kinds of neurons, which all look a bit different, but generally there will be: a cell body (where the nucleus is), dendrites coming off the cell body, and the axon, which is long and branched. Connections between neurons are made when an axon terminal comes close to the dendrites or cell body of another neuron.
A synapse is the small gap made at this connection point. Neurons can signal to each other electrically or chemically. A chemical synapse is 20 to 40 nanometers across. Chemicals called neurotransmitters are passed across the synapse for signalling.