Starting at first level monks can use their Martial Arts ability to make an unarmed strike as a bonus action (and later they get more than one attack with Flurry of Blows).
When you use the Attack action with an unarmed strike or monk weapon, you can make one unarmed strike as a bonus action. (PHB page 78)
In the section of the PHB detailing the rules of attacking (page 195 for this quote), there is a paragraph about unarmed strikes that says
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike […] On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your strength modifier.
Later on the same page, the PHB goes over the rules for two-weapon fighting (emphasis mine):
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.
Certain classes, such as fighters and rangers, get access to the fighting style Two-Weapon Fighting to get around this:
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack. (PHB page 91)
However, monks do not get this fighting style. So assuming that a monk would not have a negative modifier to their unarmed strikes, do monks not add their ability modifier to unarmed strikes used during their bonus action? Does an unarmed strike count as an attack with a light melee weapon?
Best Answer
You are overcomplicating things here.
As a monk, your unarmed strike damage is replaced with a d4 (second Martial Arts bullet point, PH78), and "[y]ou can use dexterity instead of strength for the attack and damage rolls of your unarmed strikes and monk weapons" (first bullet point), so you ignore the normal 1 + Str mod.
Also, monks are not using the two-weapon fighting rules, they're using the Martial Arts rules. Last bullet point says you can make an unarmed strike as a bonus action—which, according to the first two bullet points, is Dex + Prof bonus attack bonus, with a d4 + Dex damage roll (at 1st level).
When using the Martial Arts rules, you follow the rules for Martial Arts, not any other rules. The specific MA rules override the general rules you are citing from elsewhere.