As mathematics developed over the course of history, mathematicians needed more and more symbols to represent the numbers, functions, sets, and equations that were coming to light. Because most scholars had some understanding of Greek, the letters of the Greek alphabet were an easy choice for these symbols. Depending on the branch of mathematics or science, the Greek letter "delta" can symbolize different concepts.
Upper-case delta (Δ) often means "change" or "the change in" in mathematics. For example, if the variable "x" stands for the movement of an object, then "Δx" means "the change in movement." Scientists use this mathematical meaning of delta often in physics, chemistry, and engineering, and it appears often in word problems.
In Algebra, upper-case delta (Δ) often represents the discriminant of a polynomial equation, usually the quadratic equation. Given the quadratic ax² + bx + c, for example, the discriminant of that equation will equal b² - 4ac, and will look like this: Δ = b² - 4ac. A discriminant gives information about the quadratic's roots: depending on the value of Δ, a quadratic may have two real roots, one real root, or two complex roots.
In geometry, lower-case delta (δ) may represent an angle in any geometric shape. This is because geometry has its roots in the work of Euclid in ancient Greece, and mathematicians then marked their angles with Greek letters. Because the letters simply represent angles, knowledge of the Greek alphabet and its order isn't necessary to understand their significance in this context.
The derivative of a function is a measure of infinitesimal changes in one of its variables, and the roman letter "d" represents a derivative. Partial derivatives differ from regular derivatives in that the function has multiple variables but only one variable is considered: the other variables stay fixed. A lower-case delta (δ) represents partial derivatives, and so the partial derivative of function "f" looks like this: δf over δx.
Lower-case delta (δ) may also have a more specific function in advanced mathematics. Kronecker delta, for example, represents a relationship between two integral variables, which is 1 if the two variables are equal, and 0 if they are not. Most students of mathematics won't have to worry about these meanings for delta until their studies are very advanced.