You may make 3 attacks, but there seem to be other points of confusion
This is more or less going to be a walkthrough of the following terms: the Attack action, Two-Weapon Fighting, the Two-Weapon Fighting Style, and finally Horde Breaker.
The Attack action
The "Actions in Combat" section starts off:
When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise [...]
The first subsection, the first action presented, is the "Attack" action.
The most common action to take in combat is the Attack action, whether you are swinging a sword, firing an arrow from a bow, or brawling with your fists.
With this action, you make one melee or ranged attack. See the "Making an Attack" section for the rules that govern attacks.
This is its own action, and it is typically how you will damage an enemy. It is also the primary way you will ever make any sort of attack. Note that the Attack action, is this specific thing, it uses your action (you get one action on your turn) and it allows you to make an attack. For further reading there is the question "What does upper-case-A-Attack action vs. lower-case-a-attack mean?". There are other ways to make an attack though, for example:
The section on "Two-Weapon Fighting" states:
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.
This is simply a subsection of the "Melee Attacks" section and is available to any and all creatures. This is an option during combat and is very specific in when one an actually do it. Several conditions need to be satisfied:
- You must take the Attack action
- The attack granted by the Attack action must use a light melee weapon that you are holding in hand
- You must be holding another light melee weapon in your other hand
- You must have your bonus action available (you only get one bonus action per round and you must use it on your own turn)
This is an option during combat and it allows you to make an additional attack by expending your bonus action; one thing this does is it gives every character a way to use their bonus action.
Two-Weapon Fighting Style
The Two-Weapon Fighting Style is a Fighting Style that specifically the College of Swords Bard, the Fighter, and Ranger. In particular let's look at the Ranger's Fighting Style feature:
At 2nd level, you adopt a particular style of fighting as your specialty. Choose one of the following options.
[...] Two-Weapon Fighting
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack [...]
Ordinarily when engaging in Two-Weapon Fighting you cannot add your ability modifier to the bonus action attack's damage, but if you choose this fighting style that is exactly what it allows you to do.
Now that we've gone over the Attack action and Two-Weapon Fighting let's look at Horde Breaker:
Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.
Note that this can only be used once on your own turn and that, unlike Two-Weapon Fighting which requires you to take the Attack action, this only requires you to make a weapon attack. Also unlike Two-Weapon Fighting, Horde Breaker does not require your bonus action, in fact, it doesn't require any sort of action whatsoever. The only restrictions on it are:
- You must have made a weapon attack
- There must be another creature within 5 feet of your first target
- They must still be within range of your weapon
- Your new attack must use the same weapon as your first attack
So we've got one attack from the Attack action, one from Two-Weapon Fighting, and a third from Horde Breaker. The first uses our action, the second our bonus action, and the third requires no sort of action.