If arithmetic is only one branch of mathematics, how many other branches of mathematics are there?

How long is a piece of string? Seriously, the answer is constantly changing, from the pre-Renaissance idea that there were two fields (arithmetic and geometry) to the 2020 classification of mathematical topics used by people producing research papers to identify the field in which their work fits. It has 63 top-level groupings, each of which has multiple subcategories. These are listed here.

https://en.wikipedia.org/wiki/Mathematics_Subject_Classification#First-level_areas

There are different ways of answering this question, but to put it simply, I would break it down like this (and others can add to this list if they like):

Algebra: Subcategories of this topic include linear algebra, abstract algebra, multilinear algebra, commutative algebra, etc.

Analysis: Includes harmonic analysis, functional analysis, etc.

Topology: point-set topology, low-dimensional topology, differential topology, algebraic topology, etc.

Geometry: Euclidean geometry, non-Euclidean geometry, discrete geometry, etc.

Number theory: elementary number theory, analytic number theory, algebraic number theory, etc.

Applied math: statistics, probability theory, numerical analysis, etc.

Combinatorics: graph theory, coding theory, etc.

In addition, there are some areas that combine methods from different areas of mathematics. For example, algebraic topology takes methods from algebra to solve problems in topology. We can probably stick this under topology but that wouldn't exactly be correct. Another obstacle to creating neat divisions between branches of math is category theory, which could be thought of as existing above all of the other branches.

So in this framework, we have six branches with their varying subtopics, and some subtopics belong to one or more branches, and some, such as category, not belonging to any branch. This list could be criticized for being incomplete or grossly generalistic, but is probably sufficient for someone who is a high school student or first-year college student thinking about deeper study of math.