A linear equation is the mathematical representation of a linear regression model. The linearity refers to the fact that the parameters enter the equation in a linear fashion. This does not preclude the equation to be nonlinear in variables. In fact, some nonlinear (in variables) functional forms can be transformed in linear in parameters through simple transformations.
A simple linear regression model represents a statistical model that relates one outcome variable called dependent variable to another variable called independent variable. In the simple linear regression model it is assumed that the independent variable "causes" or "explains" the dependent variable.
Example 1. The management believes that the number of units sold can be explain (at least in part) by the number of dollars invested in advertising.
Example 2. The cost of production it is believed to depend on the number of units produced.