Where marks come from
Win or run well and your mark goes up; run poorly and it drifts down. Trainers therefore play a long chess match with the handicapper: the dream is a horse "ahead of its mark", better than the number says, carrying less weight than its ability deserves. When you hear that a horse is "well handicapped", that is the claim being made.
Reading the racecard
On a racecard the mark appears as OR (official rating). A horse rated 140 gives ten pounds to a horse rated 130. Grades of handicap (class 1 down to class 6 in Britain) group horses of similar level; festival handicaps are simply the most competitive, deepest versions of the same puzzle.
Why most racing is handicaps
Because they are the fairest way to make competitive races out of unequal horses, and competitive races are what fields, crowds and betting markets want. Roughly half of British and Irish racing is run this way. The great non-handicap races, the Gold Cup, the Derby, are "conditions" races: the best horse simply carries much the same weight as the rest and is supposed to win.