Reading the tape, or just checking the price on snapquote is a method to trade the markets. This is more of a skill rather than science, some people can predict trends in the market looking at the way prices move with volume on the snap quote window.
So what is the person reading the tape looking at? He is checking out the way prices move and how the bid/ask and total traded quantity change. Some people develop a knack to predict stock movement looking at all this data.