At its most basic, a factor is simply a number that can be multiplied to produce another number. Therefore, any number that is not prime (i.e. that has more factors than simply itself and one), has a variety of factors. For example, the number four has the factors both of (2,2) and (4,1). Larger numbers tend to have more factors; for example, the number 45 has (45,1), (15,3), and (9,5) among its factors, though this is not the case for every large number.
Variable expressions can have factors, just as numbers can. In this case, a factor would be an expression that is common to all terms in the equation and that can thus be removed and set apart. For example, in the equation x^3 + x^2 + x, each of the terms has x as a factor; this is easier to see if you write them as separate terms: (x*x*x) + (x*x) + x. In this case, we can remove one x from each term, putting it at the front of the entire expression and leaving us with x((x^2) + x).
Under some circumstances, you may be able to factor out entire expressions from equations. Using the above example (x^3 + x^2 + x), you could add coefficients divisible by four to each term and still be able to factor. For example, if the expression were (12x^3 + 8x^2 + 4x), you could not only factor x out of each term, but also 4, leaving you with 4x ((3x^2) + 2x).
Factoring becomes important in more advanced math, especially when students work through Algebra 2 problems. Here, they will often work with complicated quadratic equations and graphs, in which it is important to reduce numbers to their simplest terms in order to work with them.