Fly

Flies are enemies that buzz around the room randomly. They cannot damage Isaac at all, even by touch, making them completely harmless. However, they can turn into Pooters if they fly around Poop for too long, or come in contact with a Golden Poop.

Along with monsters like the Gapers and Spiders, Flies are one of the most commonly-encountered monsters and appear in numerous different types.

Attack Fly
Attack Flies are red and black flashing flies that will follow Isaac relentlessly until killed. Their only way of damaging Isaac is by contact, and they move rather slowly, making them easy to kill. However, they can be dangerous when found in groups, as despite their relatively slow speed, they can swarm around Isaac quickly.

Attack Flies turn into regular Flies if Isaac has. However, they take a small amount of time to convert into Flies, so it is possible for Isaac to be damaged if the player is not careful.

Ring Fly
Ring Flies are a new addition to Rebirth. They are pink flies that, upon entering the room, will cluster together into a circular formation and orbit rapidly while moving towards Isaac. They will hold formation even when you destroy all but one fly. The maximum amount of flies per formation is 10. These flies are fast and can take up a large amount of space when moving in formation, so it may be recommended to hold these on higher danger priority.
 * Ring Flies are not affected by and will continue to stay within their current circular formation.

Moter
Moters appear as two conjoined Attack Flies. They behave similarly to regular Attack Flies, but split into two Attack Flies upon being killed. Their split is very wide and unpredictable and can easily shave off half a heart. It is recommended to kill faster flies and other small monsters before it, as the two flies it spawns are more dangerous in many ways. Running between them just as they split can be a good way to dodge them, though it is also somewhat risky.
 * If Isaac has or uses a bomb, killing a Moter will not make them split into two Attack Flies.

Eternal Fly
The Eternal Fly is a white-colored fly that can be found on any floor in the game. They usually circle Corny Poop or enemies like Holy Leeches and Hangers, and will not stop circling them until their hosts are destroyed. Before that, they will deal damage to Isaac if he touches them, and they are completely immune to all forms of damage. When their hosts are destroyed, they convert into Attack Flies.
 * If Eternal Flies are circling Corny Poop and there are no other enemies around, you can leave the room at will. This is true even if they have already been converted into Attack Flies: they will simply disappear when you return into the room.

Dart Fly
Dart Flies are similar to regular Attack Flies, but attempt to mimic Isaac's movements. Their only way of damaging Isaac is by contact, and they can move fairly quickly, making them potentially dangerous when found in groups.

Swarm Fly
Swarm Flies are similar to regular Attack Flies, but appear in tightly-packed clusters. Their only way of damaging Isaac is by contact, and much like Attack Flies, although they move fairly slowly, they can be dangerous by simply outnumbering Isaac.

Hush Fly
Hush Flies are mainly spawned by Hush and behave somewhat similar to Ring Flies, flying towards Isaac in certain patterns. Unlike Ring Flies, Hush Flies can form other shapes such as polygons and can also orbit Hush.

Hush Flies have a brief period of invincibility when Hush spawns them, making them appear to have more health than they actually have.

Hush Flies are not affected by the Beelzebub transformation.

Hush Flies can also rarely spawn outside of the Hush fight in rooms.

Army Fly
Army Flies are flies introduced in The Binding of Isaac: Repentance. They are green flies that chase Isaac, moving in the cardinal directions as they do so. If there are other Army Flies in the room, they will form a single-file line while chasing Isaac.

Willo
The Willo is a fly introduced in The Binding of Isaac: Repentance.

Willos circle around Isaac, moving with him perfectly, and firing shots inwards towards him. They circle quite quickly, and can be hard to hit. Despite the seemingly unfair tactics they use, their shots are very easily avoidable by simply staying mobile, as their shots do not inherit momentum and will usually miss Isaac as long as he keeps on the move.

It's entirely viable to ignore them at first, simply moving in a small circle to dodge their attacks while dealing with other enemies first. They also take damage from fireplaces, so it's a good idea to position Isaac such that their orbit overlaps one.

Ultra Famine Fly
Ultra Famine Flies are flies spawned by Ultra Famine that charge at Isaac on a delay. They come in 3 sizes- small, medium and large.

Ultra Pestilence Fly
Ultra Pestilence Flies are flies spawned by Ultra Pestilence that swarm around him, blocking shots and dealing contact damage. They also can be spawned by the Ultra Pestilence Fly Ball in Ultra Pestilence's 2nd phase. They come in 3 sizes- small, medium and large.

Trivia

 * The Black Fly has the lowest health of any enemy in the game, tied with the Corn and Drip.
 * Eternal Flies were first added to the Wrath of the Lamb DLC for the original Binding of Isaac game.
 * There is a fly enemy (popularly known as the Ball of Flies) in the Eternal Edition of the original The Binding of Isaac that is a ball of 4 Attack Flies, resembling 2 Moters stuck to each other. This enemy does not appear in Rebirth, likely due to being added in said Eternal Edition update that was added a while after the release of Rebirth.
 * In a Q&A Reddit thread by Edmund McMillen, Edmund confirmed that the species of flies in The Binding of Isaac are house flies and botflies.