Place the peeled garlic on a cutting board. Start slicing or chopping the cloves with a chef's knife or a large, heavy, sharp knife.
As you chop it finer, add a pinch or so of kosher or sea salt if you like, and keep chopping until it reaches the consistency you like. The salt acts as an abrasive so the garlic gets minced more quickly and breaks down a bit as the knife rubs the salt into it.
Use your non-dominant hand to add some leverage to the top of the knife, which will help make the chopping go faster. Stop when the garlic is as finely minced as desired, and remember that there is already some salt mixed in, so use a lighter hand when you season the finished dish.
For very finely minced garlic, every once in a while, use the side of the knife to smear the garlic against the cutting board, then scrape it back up and keep mincing. You can even hear the salt crunching under the blade of the knife as it helps pulverize the garlic.
Notes
How to Easily Peel Garlic
Separate the cloves from the base of the garlic bulb.
Slice the very bottom of the garlic clove off the root end, exposing the garlic beneath the papery skin.
Place them on a cutting board.
Smash the garlic: Place the side of a chef's knife (or other large, heavy knife) on each clove. Use the side of your fist to carefully give a firm thwack right on the blade to crush the garlic. Slip off the papery skin.