What is a brain wave?
Just plain electricity. Really.
Whenever we do something (think, hear, see, smell, even sleep), our brain does something as well. That something can either cause what we are doing, or happen in response to us doing it. Either way, this something is the communication between neurons. Here is a neuron:

Neurons communicate with electrical and chemical signals. The technology we use picks up on that electric activity when it reaches the scalp. That is a brainwave, and here is what it looks like:

However, we aren't interested in seeing what your brainwaves do in general. We want to see what they do when something specific happens (like an "event"-hence the technique being called Event-Related Potentials[ERPs]).
I always find the following picture to be great for explaining ERPs. Basically, the woman bellow sees the word dog many times. We record her brainwaves and use an amplifier to see the voltage. We then chop up the brainwaves to only the second after she has seen the word. We average those bits out...and voila! We have an ERP!

In this case, the event is kids looking at our emotional images.We can answer questions like: Do kids show different brain reactions to happy versus sad pictures? Do they react like adults? Do shy kids respond in a very limited way? How about anxious kids, do they react more when they see a scary picture?
Image credits: EEG by Teddy Poh 2013, Neuron by http://www.artsci.wustl.edu/~alambert/chpt5_files/frame.htm, ERP by http://www.coastalneurotherapy.com/ERP.html
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