"Doneness" is "how cooked do you want it?"
It's a gradient from raw (just cut from the animal carcass) to charcoal (generally burned to an inedible mass of carbon and trace minerals). The more the meat is exposed to heat (by both temperature and time), the more cooked it gets.
A whole bunch of changes occur during the heat exposure, moisture levels change, heat activates enzymes, protein strands change shape ("toughen"), carbohydrates break down (caramelization/Maillard reaction), fats melt…
And at some midpoint between raw and charcoal, most people find they like it a certain level of cooked. Less ("it's still raw in the middle!") or more ("it is tough like boot leather all the way through!") seem unpalatable.
A number of methods have been developed to ascertain doneness, squishing it with fingertip pressure and measuring internal temperature are very common ways to get to the desired result.
If you don't know better, you want medium rare for lamb or cuts of beef, medium well for ground beef, well for pork or chicken.
Rare vs medium rare, rare basically is still raw in the middle, medium rare is partly cooked in the middle (pink, not red). Beef goes from red to pink to grey as the cooking progresses, but may brown on the outside as the maillard reactions take place.
Rest your meat after cooking, remove from the heat (out of the pan), cover loosely (cloche or foil tent or domed pan lid, not touching the meat), wait 5 minutes before cutting.