When doing data analysis, we're often looking for relationships in our data. Does one variable correlate with another? If we have more of one thing, do we have less of something else? Does, say, a person's body mass index (BMI) have a relationship to the longevity of her life? This isn't always obvious just by looking at a graph. A relationship that seems obvious to our eyes may not be significant.
Linear regression is a way of finding a linear formula that matches the relationship between an independent variable (the BMI) and a dependent variable (longevity). It also tells us how well that formula explains the variance in the data and how significant that relationship is.