You have one opportunity attack per round in DnD 5e. That’s because an opportunity attack requires “you use your reaction to make one melee attack against the provoking creature” and “when you take a reaction, you can’t take another one until the start of your next turn.” (Player’s Handbook, pages 190 + 195).
In this context of the reaction rule, “a round is measured from the start of your turn. You could take a reaction, then again after your turn starts.” (Jeremy Crawford, DnD 5e lead rules designer).
So, it is conceivable that you could make two opportunity attacks in one run-through of the initiative order, but you are still limited to one opportunity attack per personal round. For example, consider the following initiative order:
On round 1 of combat, the Goblin moves out of the Rogue’s reach, and the Rogue makes an attack of opportunity against it.
At the start of the Rogue’s turn, they get their reaction back. After the Rogue takes their turn, the Hobgoblin moves out of the Rogue’s range, and the Rogue makes an opportunity attack against it.
Technically, the Rogue made two attacks of opportunity in one run through the initiative order. But the first one happened before their turn, and the second after their turn, on a “new round” for them personally.
Note that you only have one reaction per round; not just one opportunity attack. That means that spells and abilities, like Shield, Counterspell, Riposte, Uncanny Dodge, etc., can’t be used on the same round as making an attack of opportunity in DnD 5e.
How Can I Use More Opportunity Attacks per Round in 5e?
There are two very niche and high-level ways to get more opportunity attacks per round in DnD 5e:
Cavalier Fighter. The 18th-level feature Vigilant Defender allows you to take a special reaction on every creature’s turn, which you can only use to make an opportunity attack. With this feature, you can make as many opportunity attacks as there are enemy creatures participating in the fight.
True Polymorph. A 9th-level spell (so 17th-level caster), with which you can turn into a Marilith, which has one reaction on every turn in combat. This can be used to make as many opportunity attacks per round as there are enemy creatures.