Brain Waves, or more properly termed - the human electroencephalogram (EEG) - are a physical representation of the electrical activity in your brain. Indeed, electricity plays a key role in the function of your brain. As a starting point, let's look at what a neuron is.
Neurons are the building blocks of your brain. Peripheral afferent neurons bring signals from your muscles or other sensory areas to your brain. Peripheral efferent neurons carry signals to your muscles to make them contract so you can move. Inside your brain, a vast number of interneurons - neurons connected to other neurons - give rise to all of your brains activity - from feelings of fear and hunger to the complicated decisions you make.
Neurons communicate with each other via action potentials. The video below gives a good overview of what an action potential is.
In short, the action potential is the electrical signal generated by a neuron to communicate with other neurons. When the action potential reaches the end of the axon - it causes a release of neurotransmitter (a chemical compound) that crosses the small gap between the neuron and the next neuron where it binds, or joins, with the next neuron. That process results in a small electrical potential, or voltage, called an excitatory or inhibitory post synaptic potential. If a large number of neurons are grouped perpendicular to the surface of your head then electrodes on the scalp can measure that voltage activity, the excitatory or inhibitory post synaptic potentials - or the EEG / "brain waves".
For comparison, think of ECG or the electrical activity of your heart. Electrodes places on your chest can detect the electrical activity that is needed to contract your heart. The "flat line" we all fear is the result of there being no more electrical activity to contract the heart muscle.
EEG is similar in a sense - it is the electrical activity of millions of neurons firing in your brain. For example, when you see an image, think of a friend, or get hungry millions of neurons "fire" to create the pattern of thought. We can measure this neural activity with EEG.