Getting started, it is important to recognize the difference between expressions and statements. Expressions are combinations of operations and numbers, some of which may be denoted by variables. They are simply instructions to do certain things mathematically, which may or may not be possible to complete at first. Expressions can be simplified or rewritten using appropriate rules, and evaluated by choosing constants to replace any variables in the expression.
Statements are constructions that involve a comparison of the values between expressions. The most common is an equation, but inequailties and other comparators can be used to create statements. The goal of an equation (and other statements) is to solve them — meaning to find all values that you can replace any variables with so the statement is true. This may yield any number of solutions, from none to all numbers (for statements with one variable).
While equations with one variable are often used to set up and solve applications, equations can also be used to describes relationships between two variable values. These equations in two variables are typically written with a single variable on one side and the remaining expression on the other. In many cases, the pairs of numbers that satisfy the equation is called a function, and is a topic of much study in algebra.