Assay is like a test for grown-up stuff, sort of like how you have tests in school to see how well you understand things. But instead of testing your knowledge, assay is a test to see how much of something is in something else.
For example, imagine you have a big bowl of M&M's and you want to know how many red ones are in there. You could dump them all out and start counting, but that would take a really long time! Instead, you could do an assay using a machine that can quickly sort the M&M's by color and count how many of each color there are. This is a quicker and more accurate way of finding out how many red M&M's there are.
Scientists use assay all the time to learn more about different things. They might use an assay to see how much of a certain chemical is in a sample of water, or how many cells are in a tissue sample. It's a way of figuring out exactly what something is made up of, so that we can understand it better.