The rules for jumping in D&D aren't overly complex, but I've always been slightly annoyed at their confusing layout. With an unnecessary mix of squares and feet as distance units, sections that depend on a character's height, and examples that aren't clear enough, jumps could eventually be figured out, but always seemed to take me more time than they should.
I understand that space is at a premium in a rule book like Player's Handbook, but I think a few tables with the DCs for the most common jump scenarios, coupled with graphic examples where you could clearly view distances jumped, moved, and cleared vertically, would go a long way towards solving this problem. So that's what I have done. In this post, you'll find (hopefully intuitive) examples for all kind of long jumps and high jumps with all distances converted to squares.
1- Long Jump
Most of the time, when a character makes a jump, it will be a long one - they are useful for clearing pits and avoiding certain types of difficult terrain, among other useful applications. The DCs are usually very manageable (even without running starts!) unless you're crossing great distances, though it gets significantly trickier when you add vertical obstacles to the mix. Note that, when you clear N squares with a jump, you're actually moving N+1, since you also need to enter the square you land on.
Fig. 1 : Long jump without running start.
Fig. 2: Easy (DC 25 or less) long jump after a running start:
Fig. 3: Hard (DC 30 or more) long jump with running start.
2- High Jump
High jumps are relatively rare, but come up whenever you want to grab a ledge or a rope, among other things. They are slightly less efficient for clearing vertical distances than long jumps, in terms of required DC, though they also require far fewer squares of movement.
PHB provides rules for jumping a distance in feet, which have been translated here to full squares. In case you need more granularity, I'd recommend using fractions of square - since DC is directly proportional to the distance cleared, if you needed to jump across 1/2 square you'd need a check of 50/2=25 without a running start, or 25/2=13, with one. Also, remember that unlike long jumps, vertical distance cleared this way adds to your total distance moved.
Fig. 4: High Jump
When trying to reach something above you by jumping, you can take into account your character's height. According to PHB, your vertical reach this way is equal to your height plus 1/3 - converting this to squares, and playing around with the numbers, we find that a character's reach will increase by 1 full (5 feet) square for each 3.75 feet of height.
In practice, most PCs will have a reach between 1 and 2 squares. Halflings sometimes fall below the 3.75 mark, and gnomes do so most of the time. On the other end of the scale, Goliath are the only playable race that routinely goes beyond 7.5 feet. For simplicity's sake, I'd recommend finding your PC's reach in squares, rounded down (i.e, 1 square 99% of the time) and sticking to that, though if you want a more rigorous approach, it may be a good idea to write down vertical reach translated to squares (height/3.75) in your character sheet. For example, a 6 feet-tall human would have a reach of 1.6 squares.
Fig. 5: Vertical jumping reach.