I am taking my first year in my bachelor of science, hoping to major in Environmental sciences. I haven't done any math other than your basics, such as dividing up grocery bills or trying to figure out my height in millimeters (1753mm by the way), for years. This is turning out to be harder than expected!
When looking up the exact definition for analytical geometry I came across one that summed it up pretty nicely, “analytical geometry is just a fancy name for graphing”. It is geometry (basically about shapes and their properties), using coordinates and algebra. Renee Descartes built on previous methods similar to analytical geometry and was able to use coordinates (numbers describing points) to express geometric relations/shapes in algebraic equations, becoming the “father” of analytical geometry. Using the most common coordinate system called the Cartesian Coordinate system, where every point has numbers associated with it, one representing the x (horizontal position) coordinate and the other the Y (vertical position) coordinate. Any equation involving coordinates on a plain gives you the solution set for the equation representing the line formed from those points (coordinates). Depending on the equation, the variable and the coordinates, it will tell you the basic picture/type of graph you will get (straight, curved, positive or negative slope). Using these coordinates and numbers you can solve for unknown lengths, angles, vectors, intersecting points… ex: linear equation ax+bx+c = 0, represents a straight line, where a,b, and c are constant numbers, by this you can than plug them into algebraic problems/formulas. Why is it important? It helped permit the evolution of other modern mathematics such as calculus by Newton and Leibniz (although I do not know who the latter person is).