When doing data analysis, often we look for relationships in our data. Does one variable affect 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 his/her longevity? 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.