Spells are a controlled manifestation of magic that supernaturally affects the world. Although this is the most basic ability of a wizard, there are multiple categories, types, and branches of this.
Please note that although a spell has "spell" in the name it does not mean that it does not have a classification and can be just a simplified name. For example, the Stunning Spell. Although the name is The Stunning Spell, its technical name is the Stupefying Charm and is classified as a charm.
There are two categories of spells- Non-Dark and Dark spells. The difference between the two stems from their effects, and how they're cast. While all spells rely on concentration to be cast, Dark spells require negative emotions to give them power. This and their dark effects is why they are referred to as dark.
There are two types of Non-dark spells- Transfiguration, and Charm. Counter-spell and Healing spell both being types of spells is a misconception. They are what are referred to as classifications.
Classifications further explain what the spell does. They have overlap with every type of spell and aren't considered different enough to be considered spell types. Although just because they aren't considered different types, that doesn't mean they aren't as equally important. Counter-spells are extremely important in Auror work as well as many other professions and have multiple different types. Healing spells are mainly taught at post-graduate Healer schools such as Horncreast.
Transfiguration in its most basic definition is the alteration of the object's form or appearance. Transfiguration has its limitations, however. It is easier to transfigure an object to another when they share similar properties. Mass and size also play a role in the ease of the spell. You cannot revive the dead using transfiguration. Although there are more limitations this is just a brief overview and you can find a more detailed explanation in A Beginner's Guide to Transfiguration. One of the reasons why it's an entire class is because it has four branches- Transformation, Transformation, Vanishment, and Conjuration. (These branches are listed in ascending order of difficulty) See a more detailed explanation of these branches as well as sub-branches on chapter 13, page 57.
There is some debate however on this. Some say that due to the overlap with some charms that they should be considered Classifications but this has not reached a general consensus and is still considered branches of Transfiguration.
Charms, or Enchantments, alter an object's inherent qualities. How Charms differ from Transfiguration is that Charms alter what the object does as opposed to what the object is. Charms also, typically, last longer than Transfiguration. The time in which a charm lasts depends on the charm; some last until death, some last until the caster stops supplying it with magical energy (Later renamed to Mana by Dumbledore).
Although the Jinx, Hex, and Curse are all considered different types of Dark spells, in reality, it is mostly a difference in power and all can be considered Dark Charms.
Jinxes are minor dark magic. Their effects are considered irritating but amusing, almost playful, and of minor inconvenience to the target. Jinxes can only be maintained as long as the caster keeps eye contact. The Counter-spells for blocking or removing Jinxes are counter-jinx, counter-curse. They can also be prevented with Anti-Jinx.
A Hex consistently negatively affects the object. Major inconvenience to the target. They can be countered using the Counter-spell types Hex-Breaker or Hex Zapper.
The final Dark Charm and the worst kind of dark magic are Curse. Curses, unlike most spells, are inherent to nature. For example, the act of drinking unicorn blood gives a Curse. The three worst Curses are Avada Kedavra, Imperius, and Cruciatus. These three are called the Unforgivable Curses and will earn the caster a life sentence to Azkaban if used against a human life.
When writing chapter 23 I reference Spell Types and I had a sper of creativity to write this, hope you liked it. I wrote it as sort of an experpt from a textbook.