Fertilizer is always a contentious issue, because a lot of businesses sell fertilizer in myriad forms. These sellers will all tell you that you need more and more, which is a load of… uh… fertilizer.
Fertilizer is basically a mixture of essential elements. Nitrogen (N), Phosphorus (P), and Potassium (K) are the big three, and any mixture of fertilizer which uses a number sequence like 20-20-20 is referring to the percentage of of N P K within that container. There are about a dozen other required elements, called micro nutrients, which are needed sometimes in very small amounts.
All soils contain some or most of these elements. All plants require most of these elements in greater or lesser degrees. When a plant is grown in soil that is deficient in an element that it needs, that plant or crop will not thrive. Careful soil analysis can be very important especially if we are talking about a commercial crop grown in the thousands of square miles.
Some plants thrive and respond better to fertilizer than others. Annuals and plants that are stressed in their growing conditions usually respond well to fertilizer. Examples would be the agricultural crop mentioned above; another example is lawns. Annual flowers and vegetables respond well to regular fertilizing and the application of organic compost, which contains a lot of the elements a fertilizer would contain, with the added benefit of organic plant material for the soil.