There are many ways of moving without triggering Opportunity Attacks.
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
You can avoid provoking an opportunity attack by taking the Disengage action. You also don’t provoke an opportunity attack when you Teleport or when someone or something moves you without using your Movement, action, or reaction.
Like the other answer stated, you can use the Disengage action (instead of attacking, etc).
If you take the Disengage action, your Movement doesn’t provoke opportunity attacks for the rest of the turn.
Disengage as a Bonus Action
Some classes (like Rogue) can do these actions as a Bonus Action, so they can attack and then disengage.
Spells like Teleport or Misty Step move you without triggering opportunity attacks.
You teleport up to 30 feet to an unoccupied space that you can see.
Moving within reach
Another way of moving, albeit more restricted, is to move around your enemy without leaving his range (and thus not giving him an opportunity attack). This can be useful if you're giving your enemy cover from your allies in the back, and you just rotate to his side.
Purposely expending enemy reactions
Another way you can move is to wait until he has expended his reaction. I've seen a ranger purposefully use his animal companion in such a way as to tank opportunity attacks so that enemies no longer have reactions and can't do Opportunity Attacks or Counterspells.
However, what you might be most interested in is to push and shove your opponents, with the special attack Shove.
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. You make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use).
If you win the contest, you either knock the target prone or push it 5 feet away from you.
Once he is away from you, you can move away (assuming he doesn't have Reach over 5ft).
Spells that move enemies
You can also use spells that move enemies away, such as Thunderwave or Dissonant Whispers (thanks to Marq).
ThunderWave - A wave of thunderous force sweeps out from you. Each creature in a 15-foot cube originating from you must make a Constitution saving throw. On a failed save, a creature takes 2d8 thunder damage and is pushed 10 feet away from you.
Dissonant Whispers - On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you.
Enemies can only make these attacks if they can see you. Spells like Invisibility solve this.
A creature you touch becomes invisible until the spell ends.
Any condition that affects sight, attacks, or reactions will influence this. The list includes
Numerous spells, such as Fog Cloud, Sleep, etc, will cause these conditions.
There are many ways to hinder enemy attacks. Ones that come to mind are disarming enemies, polymorphing them into harmless creatures, applying conditions that cause disadvantage, or receiving buffs that cause disadvantage to attackers. Keep in mind that even without weapons or attacks, any creature can perform an unarmed strike.
Grapple an enemy and drag him around with you. You can move him out of reach of your allies (and so they can move about), and you can move around with him without triggering his Opportunity Attacks. With some luck, you can just rotate around and move him away from your allies without having to move yourself and taking other nearby enemies' attacks. Also remember that you can Grapple 2 separate targets and still perform Unarmed Strikes. With the feat, after grappling someone, you can grapple them again to apply the Restrained condition (thanks to Doval).
If a friend is in trouble, Grapple your ally and move him away (forced movement does not trigger OAs). As a house-rule (thanks to Shane), I as a DM would argue that since your ally willingly lets you move him, it is not forced movement and thus triggers Opportunity Attacks anyway, also since Grapple isn't really intended for allies, but... that would only be my house-rule, your DM will likely have a different interpretation.