Although it has been proposed that there are anything between 50 and 500 different types of neurons in our brain, they are mostly just specialized cells based upon the basic neuron. The basic neuron consists of synapses, the soma, the axon and dendrites. Synapses are connections between neurons - they are not physical connections, but miniscule gaps that allow electric signals to jump across from neuron to neuron. These electrical signals are then passed across to the soma which performs some operation and sends out its own electrical signal to the axon. The axon then distributes this signal to dendrites.
Dendrites carry the signals out to the various synapses, and the cycle repeats.
Just as there is a basic biological neuron, there is basic artificial neuron. Each neuron has a certain number of inputs, each of which have a weight assigned to them. The weights simply are an indication of how 'important' the incoming signal for that input is. The net value of the neuron is then calculated - the net is simply the weighted sum, the sum of all the inputs multiplied by their specific weight. The output is then fed into all the neurons it is connected to.