this hasn't been math in a long while

Real numbers are any number on the number line. Most people deal with real numbers, or rather the small subset of real numbers (like Natural numbers, integers, or rational numbers). It is represented by the letter R with a fancy font, for obvious reasons.

Complex numbers are numbers on the complex plane, which is usually defined as being a real component and an 'imaginary' component. The square root of negative one is impossible to resolve with real numbers, so it is considered to be orthogonal to the number line. They are very useful when dealing with electricity, quantum physics, and frequency phases. It is represented by a fancy letter C, again for Obvious reasons.

Quaternions are what you get when you take two Complex numbers that are orthogonal to one another (or 4 dimensions of Real numbers). Quaterions are a highly advanced topic, even more so than complex numbers. But they are VERY good at representing ways to rotate or manipulate 3d objects, and have a lot of use in computer graphics, physics simulation, and fluid dynamics. They are represented by the fancy letter H because the fancy letter Q was already used for Rational Numbers (who had to take Q because R was already used by Real numbers). H stands for Hamilton, a mathematician who worked with them.

Octonions are what you get when you take two Quaternions that are orthogonal to one another (or 4 Complex Numbers, or 8 Real numbers). They are extremely niche, but are mathematically consistent in the same way as Quaternions are (with a few lost properties along the way). They are represented by a fancy O.

Sedenions are what you get when you take two Octonions that are orthogonal to one another. You can keep going doubling and doubling over and over again, but at this point things have already gotten silly.