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Spot reduction is the idea that body fat can be reduced in particular problem areas, such as the glutes or lovehandles, by repeatedly performing exercises that specifically target those areas. Like all broscience, the concept is lacking in anything that resembles non-trivial evidence.
Why it doesn't work
The human body, like the bodies of most other mammals, stores unused energy from sugars and complex carbohydrates as fat. This is an evolved quality of mammals—fat serves as an insulator in times of cold temperature and as reserve energy in times of famine. That fat is stored throughout the body, below the skin and above the muscles (a positioning which also helps to lubricate muscle movement). Fat may bulk up in certain areas where there is more room (such as the outer thighs, the neck and breasts), but it is more-or-less uniform. If fat were to go "straight to my thighs!" as many people might complain about, there would be many people out there who are svelte and trim in some parts of their bodies and horribly distended around their "problem areas". Every individual is different; your experience may differ slightly, but generally the storage is uniform.
When an individual starts a weight loss routine, ideally through changing to more healthful eating habits and a regular exercise regimen as prescribed by a qualified dietitian, fat loss occurs the same way it was put on— uniformly about the body. Those certain areas that have more fat storage by nature of their location may not reduce as quickly as the other areas, which might suggest spot reduction is occurring, when in fact the fat is simply being reconverted into energy at the points most accessible to blood vessels. Exercise simply increases the fat "burning".
Let's put it another way. Assume spot reduction works and it's the way fat is removed from the body. That means someone who is looking to lose fat would have to work out every "spot" of their body in order to experience results. Otherwise, a person whose only form of exercise is daily walks in the park would only lose fat around their legs and hips but never in their torso or neck area. Since it is known that an exercise routine will eventually reduce body fat to healthy levels throughout the body if an exercise regimen is followed, one can conclude through reductio ad absurdum that fat loss doesn't work that way.
What actually does happen
While limiting exercise to "targeted" areas of the body will reduce fat all over, it will only increase muscle mass in the areas which are exercised (in weight-lifting/body-building circles this is referred to as isolation). Muscles that get larger leave less room for fat, which will cause fat to be compressed in those areas but won't actually increase the fat reduction above and beyond the normal loss attributed to the exercise. Focusing the exercises to the detriment of other muscles groups can cause abnormal body shapes or unbalanced muscle coordination, leading to problems such as being funny looking (e.g., "never skip leg day") or injury (e.g., just because someone has massive biceps doesn't mean their triceps can handle a strenuous exercise).