Spellcasters such as wizards and clerics, as well as many monsters, have access to spells and can use them to great effect in combat. Each spell has a casting time, which specifies whether the caster must use an action, a reaction, minutes, or even hours to cast the spell.
When you take the Magic Action, you are utilizing magic by casting a spell that has a casting time of an action or by using a feature or Magic Item that requires a Magic Action to be activated. The specific rules of the Spell, Feature, or Magic Item being used will specify which type of action--be it an Action, Bonus Action, or Reaction--that the Magic Action is required to utilize in it's activation.
If you cast a spell that has a casting time of 1 minute or longer, you must take the Magic Action on each turn of that casting, and you must maintain Concentration while you do so. If your Concentration is broken, the spell fails, but you don’t expend a spell slot.
For a full rules breakdown on Casting a Spell, see here.