Companion is a deckbuilding restriction printed on certain creatures. If your deck (and starting hand/library) meets that creature's specific condition, you may reveal it from outside the game and play with it as a sort of "bonus card." Once per game, you pay 3 generic mana to put the Companion into your hand, after which you can cast it normally from there.
The most common mistake: paying the 3 mana doesn't cast the creature—it only moves it to your hand. You still have to pay its mana cost separately afterward. Also, the deckbuilding condition must be satisfied by your entire deck, not just at one moment.
In Commander, the Companion lives in your command zone alongside your commander and starts the game outside your 99. It effectively gives you reliable card advantage, though the restrictions (like Jegantha's no-duplicate-pips rule) can heavily shape your build.