Regression analysis is a set of statistical tools used to identify trends in data, and specifically to quantify the relationship between a dependent variable and an independent variable (or multiple independent variables). For example, you might survey people on, for example, how much education they have and what their income is and, looking at the data, believe that there is a relationship between the two; to quantify that relationship, you can use regression analysis. Regression analysis provides an equation that can help you make predictions. You could use such an equation to answer, for example, how much someone with 17 years of education can expect to earn.
The most common regression analysis technique is linear regression. In linear regression, the relationship between variables is described by a straight line. Straight lines are easy to understand, and many phenomena are related linearly. If a relationship between two phenomena isn't linear, transformations can be used to transform that relationship into a linear one.