"Inside the leather" is also the inexact measurement (because not all putters are the same length) used to determine whether a putt qualifies as a "gimmie." If a group of golfers is using gimmies, then a golfer whose ball is inside the leather will get to pick up without holing out (obviously, this is something that can only be done in casual games between friends, and by agreement between those friends - gimmies aren't allowed under the rules).
To measure "inside the leather," place the putter clubhead inside the cup on the green. Lay the putter flat on the putting surface, extending back toward the ball. If the ball is between the cup and the bottom of the grip (i.e., if the ball lies next to the shaft part of the putter), the putt is said to be "inside the leather" and, therefore, within the gimmie distance. (Be careful not to damage the edges of the hole when doing this.)
A couple of notes: 1. Don't try putting a long putter in your bag and then claiming your ball is inside the leather when it's four feet from the cup. Your buddies won't let you get away with that. "Inside the leather" can only be cited with conventional putters (most of which are 33 to 36 inches in shaft length).
2. When the term first came into usage, it referred only to the grip itself; a ball was inside the leather only if it was closer to the hole than the length of the grip on the putter. Over time, however, the meaning (and measurement) expanded to what is cited above.
"Inside the leather" can be applied to any ball that is very close to the hole, as a descriptive phrase: "How long is your putt?" "It's inside the leather."