This video is intended to be humorous, but it does show what it means to 'skip':
On the other hand (... or foot ... haha), a leap is when you take off with both feet leaving the ground and then land some distance away. That distance is variable, because you can leap over a sleeping dog, or you can leap off a building (but please, don't) -- so it depends on the circumstances.
'Leap' and 'jump' can usually be used interchangeably -- you could have jumped over that dog or jumped off that building -- but a 'leap' is a bit more dramatic than a 'jump'. If you jumped out of bed this morning, you must have slept well; but if you opened your eyes and saw a big bug on the sheet, you would have leaped out of bed.
Of course, these words are also used in contexts that don't involve actual physical movement. Examples ...
> He said to his girlfriend, "Just because you saw me kissing that girl, don't leap to conclusions." [People often 'jump to conclusions', but in this case, it would have definitely been a 'leap'.]
> Who wouldn't leap at the chance to work for that company!
> He told his girlfriend about the great restaurant he went to on his trip, but he skipped the part about the beautiful waitress.
> I know this next meeting will be boring, but the boss will be there so don't skip it.
Well, I may have skipped over other good examples, but you get the idea.