The key point in binary linear programming is to optimize a certain function. The function you want to optimize is called the objective function. This function is always linear but can have an unlimited number of variables. For example, Z = 2x + 4y + 3z is an objective function. The optimization of the objective function yields the largest or smallest value for that function, depending on the situation, such as whether the user wants to maximize or minimize the function.
Optimizing a function is easy and should not be a mathematical field in itself. The reason binary linear programming is a mathematical field is that the form of optimization is not like that of normal mathematical optimization. In binary linear programming problems, the objective function is subject to a set of constraints. That is, there are other functions limiting how the objective function can grow or shrink. These constraints are in the form of inequalities. One example of a constraint is -2x + 6y -- 3z > 2.
The reason binary linear programming includes the word "binary" is because of the additional set of constraints, which is one of the variables themselves. This constraint is the same for all binary linear programming problems. Namely, all the variables (x, y, z and so on) must be binary. That is, these variables can only take two possible values: 0 or 1.
Binary linear programming is well-studied because of its applications. You might wonder when would there be a situation in which all the variables of a function can only be binary? In fact there are many such situations. The best way to understand what a binary linear programming represents is to think of the variables as "yes/no" or "on/off" decisions. When a variable equals one, there is a decision to include that variable. For example, when gathering a team together from a subject pool, you have two decisions for each subject: Take or reject. Each of these decisions will affect the efficiency of your group. This type of problem can be modeled as a binary linear programming problem.