Death from Above
A quick glance at the variety of aerial trajectories often leads to the misconception that KOF lacks a solid ground game and instead primarily revolves around short hops and straight high/low rushdown. However each commitment to a jump means forfeiting the option to block just as in more traditional 2D Fighting Games, ergo the aerial player is completely vulnerable to anti-airs or attacks that control certain portions of the aerial space (e.g. projectiles, K's Ein Trigger and Second Shell, Mai's Weak Ryuuenbu). Repeated hops can be anti-aired with a quick light attack at the least (which trigger a hit reset, giving the grounded player enough time to set up a meaty, tick throw, to reestablish space, or reverse the momentum) while full jumps are as easily punishable as in other fighters. Even without anti-airing the defending player can incorporate defensive option selects and techniques to create a rigid wall of defense that is not easily topple-able.
Smart hops and jumps, then, rely on the spacing between players and which jumping attack to use. Approaching with a hyper hop combined with the exact spacing to hit with the tip of a long-reaching jumping attack creates a difficult situation, and even if the opponent performs a 'DP' type invulnerable move the player could always do an empty hyper hop so that the opponent's anti-air whiffs because it cannot catch an outstretched limb. If the hopping player's spacing is incorrect due to a miscalculation or from the opponent walking or running forward it becomes easier for the grounded player to anti-air by getting underneath the opponent for a deep anti-air or tripguard meaty which can turn into a combo.
Due to the number of possible jump arcs, KOF allows players to attack from many angles that aren't quite coverable in Street Fighter. These types of approaches function offensively (as mentioned above) and defensively; mastery of KOF's mechanics require knowing which angles both players can choose to control and knowing how to counter each setup.
A visual guide to the different types of jumps and options off of each can be viewed here.
Short Hops
Oftentimes referred to as just a "hop" and abbreviated as h as a measure of shorthand, short hops are performed by quickly pressing [ 
][ 
] or [ 
] and then inputting a neutral or any non-upward motion. Performing a short hop can be difficult at first but being able to select from any jump option at a moment's notice is absolutely vital to KOF. One trick is to quickly move from an upward direction to down [ ]~[ 
] in order to avoid holding [ ] too long and causing a jump.
Short hops are fast and possibly the least vulnerable of all jump types (the hopping player moves forward under 1/3 of the screen's length) which make them difficult to react to when under pressure. Using air-to-ground attacks from short hops is a quick way to keep the opponent blocking while gaining frame advantage afterward. Once used to seeing short hops, they become easier to option select block and then anti-air when an opening presents itself.
Short hops are great for baiting and then punishing sweeps. This forms part of the basic RPS system of KOF where hops beat low attacks, standing attacks defeat hops, and low attacks win against standing attacks.
Hyper Hops
Hyper hops (hh) are performed by first pressing [ ] ~ [ or ] and then releasing to a non-upward directional input. They can also be done when running by trying to perform the input for a short hop. Note that hyper hops cannot be done neutrally upward [ ].
Hyper hops are even faster than their short variants and travel further (under 1/2 the screen length). This makes them suited for quickly closing the distance or getting back in on the opponent with a jumping attack. One weakness is that a low hop arc allows normals and specials that dominate the 'short hop space' to naturally anti-air hyperhops. Some projectiles can be hyper hopped over from proper distances with tight timing but it is still possible to end up hopping into a projectile than to avoid one. Against most non-grounded projectiles it's easier to jump to avoid the attack, but opting for a jump puts the player at a greater risk of being anti-aired.
Each character has a specific hop height; some can hop over standing opponents while other require the opponent to be crouching to cross over. Many common crossup setups are done by attacking with one or two light attacks and then short or hyper hopping at the opponent with a crossup attack, though the success of such attempts can vary depending on if the opponent crouches or stands when blocking.
Jumps
The option that should be the most familiar with most fighting game players. Jumps (shortened as j) in KOF are performed the same way as in any game: just hold [ or ].
Jumps reach a high vertical height which suits them for maneuvering over large specials or challenging the other player in the air. Some ambiguous crossups only function from a full jump such as Maxima's Jumping 
bodysplash which can't be used from a hop.
Jumping carries a longer aerial hangtime which gives the other player more substantial time to anti-air or air-to-air. Additionally, a normal jump covers the same horizontal distance as a hyper hop. Jumps end up getting punished more often than either hop, and so much like in any fighting game the player should have a set reason or strategy in mind before jumping heedlessly.
Super Jumps
Super jumps (abbreviated as sj) are done by pressing [ ] ~ [ or ]. Players cannot super jump neutrally upward [ ].
A super jump travels a bit further than 1/2 of the screen length. Combined with the widescreen ratio this creates more safety from fullscreen against being jumped on. A superjump travels faster and further than a standard jump although tall vertical height of the jump arc makes this the easiest of all air options to anti-air.
Although both jumping options are riskier than hops, players can use the same tight spacing principles to tip the opponent with an air-to-ground attack or come in from an angle that the opponent's character has trouble defending against.