Everything I am about to tell you is more than you need to know about lightsaber fighting styles. This is also going off a mix of pre-Disney and post-Disney lore. Meaning how much is still canon is up to the mouse. I will do my best to stick with things that they cannot change.
In the original styles of lightsaber combat, there are seven complete forms. Some have subvarients that are designed for an alternate lightsaber, such as dual wielding or a staff. However, generally speaking, when asked there are only seven in the time of the Grand Republic.
Technically there is a form called form Zero. It is just having a lightsaber present, both as a mark of as your station as a Jedi and someone who can use it to settle disputes. Arguably it is the form a master Jedi might take to focus on the use of force powers. This, however, is not really considered a form.
Form 1 is called Shii-Cho. This is the most basic form of lightsaber combat. It has no real strength nor any weaknesses. Most Jedi are taught this one when they are first introduced to lightsaber combat. It teaches them to protect themselves, and their weapon from people trying to take it or turn it off with the force. Very few go on to master this style or use it exclusively.
Form 2 is called Makashi. This is the style that Count Dooku used and is designed as a style for fencing. Designed almost exclusively for anti lightsaber combat and saw its greatest use during the Sith Wars. During the time of the Grand Republic only a handful ever learned it and even less mastered it. Count Dooku and Mace Windu being two of the few. It is considered weakest at fighting an opponent with a blaster or ranged weaponry.
Form 3 is called Soresu. Obi-Wan’s second fighting style. It is highly defensive and designed for efficiency of movement. So that you can wear down and tire your opponent out. Giving yourself an opening. It excels at dealing with blaster fire, but when fighting someone with brute power or someone using Makashi their circular defensive techniques might fail.
Form 4 is called Ataru. Yoda, Young Obi-Wan, and Qui-Gon Jinn used this form. It is highly acrobatic and relies on the user to use the force to augment their movements. The goal is to overpower your opponent with rapid movements and attacks. All hoping to slip through their defenses. Its weakness is that it rapidly drains someone’s stamina. This is and the cramped fighting space is what led to Quigon’s demise.
Form 5 is called Djem. The style that Darth Vader and Luke use. It has little finesse in its attacks. Instead, you put your power behind each strike. This requires that the person doing the attacks have a natural higher strength level than their opponent. If a fight between two Djem So users comes about, it comes down to a fight of strength and endurance.
Form 6 is Niman. It is a… I’m going to call it bad attempt for someone to try and combine all the other styles. It has no overall strength and weakness. Someone mastering this style would have to learn quite a bit of the other styles just to become a Niman Master. This style also has the most branch offs. With dual wielding and staff fighting coming from this style…. Usually.
To most Jedi, this is where the list stops. For the Sith there is a 7th form called Joyu. It is an extremely fast paced and aggressive fighting style. The point of the attacks is that they are so rapid you are never sure how many blades the wielder is using. There are versions that use a staff (Darth Maul) and there are some that use dual blades (Palpatine/Darth Sideous).
For the Jedi, there is a style called Vaapad. It was created by Mace Windu and he is the only master of it. The style was based on the ‘incomplete’ version of Joyu and given the name Vaapad, both names coming from the same creature. A creature that attacks you with its tentacles so fast you never know how many it has until it is dead. Hence the name of the style.
The major difference between Joyu and Vaapad is the source of its strength. Both require a deep investment in training. You have to master all other forms. Not just know them, but to master them. Then you must draw on anger and aggression, plus usually a love of combat. For a Sith, they can draw on their own and that is why Joyu was always complete. For the Jedi, it required an odd twist.
Mace Windu admitted his own dark side in his natural desire for combat. He loved to fight and that is what allowed him to become the greatest duelist of his time. Only Yoda was considered his possible equal. The problem was that if he continued to draw on that, he could fall himself. Creating Vaapad was his answer.
Instead of drawing on his own darkness, he would draw on his opponents. Using their own energy and aggression to augment his own. Finding his own form of serenity in the center of the tornado of attacks. It is this fine line between light side and the dark side that has destroyed anyone else who has tried to master Vaapad. Also why he no longer bothers to teach it.
This is obviously not a comprehensive list of styles. Just some of the background lore of the universe.