Math notes / Overview of the areas
This is more for overview of my own than for teaching or exercise.
Overview of the areas
Other data analysis, data summarization, learning

This article/section is a stub — probably a pile of halfsorted notes, is not wellchecked so may have incorrect bits. (Feel free to ignore, fix, or tell me) 
Disclaimer: There seems to be no singular definitive categorization. There are many ones like this, this is my own.
Arithmetic is application of basic operations on numbers, the stuff useful for daytoday use, and taught for that.
Terms like elementary mathematics often mean "things common to many mathematical fields, that we'll teach people once they become comfortable with basic arithmetic". It may be less daytoday, but a good basis for things you may run into at some point in any remotely technical job.
Set theory and combinatorics are usually tacked onto other courses. Both have basic parts that can be an extension of elementary mathematics, useful for things like probability, and thinking about conditions. Both also have more abstract extremes that are primarily interesting to mathematic completism.
Geometry concerns itself with shapes, sizes, relative positions, orientations and such, basic transformations applied to points and shapes, and such.
It is taught for its general usefulness in various fields, largely in euclidean space, though geometry courses tend to briefly go into polar coordinates, and sometimes basic cases of curves and manifolds.
Parts of geometry cross over into algebra, and into other fields.
Algebra studies structure, relation, and quantity, abstractly as well as in (relatively) discrete applications. Quoth wikipedia: "Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics."
Includes concepts such as symbols and variables, and studies aspects of operators, sets, and such.
Often split into
 basic algebra
 key words: variables, expressions, polynomial equations (linear, quadratic, etc.), factorization, root determination, and such
 linear algebra
 studies linear spaces (a.k.a. vector spaces) and linear functions (a.k.a. linear maps, linear transforms), and systems of linear equations.
 key words: basis, matrix, linear transforms
 Linear algebra is frequently applied in data analyses, where the assumption of linearity is not a problem
 abstract algebra
 properties of functions, operations, and such
Precalculus may be taught in secondary school and seems to exist to ease people into calculus, as more technical studies end up needing at least a decent grasp of calculus. Precalculus is often a "let's ensure we're at the same level"style review of algebra and trigonometry in more depth, and can include a little analytic geometry.
Calculus studies infinite series, limits, derivatives, integrals, and such. Some of this is taught in the higher levels of secondary school, depending on where you live. Various university studies require it, so will devote one or two firstyear courses to making sure everyone has a basic understanding.
Analysis refers more specifically to the theory and application of limits (...of sequences and of functions), differentiation, integration and measure, infinite series, and analytic functions, certain types of approximation, and often for both real numbered and complex numbered spaces/variables.
Analysis is in calculus, sometimes considered part of calculus, sometimes a more abstract continuation of it. Analysis includes some things we wouldn't call calculus, and calculus can be considered a form of analysis. You can just consider them as covering similar ground.
Also:
Number theory is a branch of (pure) mathematics math that deals with any possible properties of (positive) whole numbers  often numbers under specific functions.
As such, it crosses over into areas such as algebra, trigonometry, and others, but is also
Discrete mathematics refers not to an area, but the property of not requiring the sense of continuity (such as real numbers).
So it includes integer arithmetic, combinatorics, group theory, formal languages, (finite) graph theory, countable sets, and such.