A diet is a regimen of nutrient consumption. When somebody says they're on a diet, they are typically trying to achieve a desired result such as losing weight or building muscle.
 An actual diet
A diet is just what you eat. Colloquially, it means when somebody is purposely going out of their way to eat something different with a desired effect. The three major types of diet are:
- Weight-loss diet: Reduction of calorific intake for the purpose of burning off the stored fat on your person. This can be done simply by eating less.
- Muscle-building diet: An increase in calorific content and protein for the purpose of helping to build muscle density on a person. Simply eating more meat can be enough.
- A supposedly "healthy" diet to reduce the likelihood of certain diseases such as cancer or heart disease. Weight loss may also be a goal here. Many vegetarians, for example, think that vegetarianism is healthier. The scientific evidence on this one is mixed, to say the least - correlation is not causation, and just because some healthy people are vegetarians, does not mean that becoming a strict vegetarian will necessarily make you healthier. However, reducing or eliminating processed meats from your diet may reduce your risk of certain types of cancer.
Typically just changing what you eat isn't going to be enough. A diet combined with exercise typically is more effective at achieving any result.
Some people adopt a vegetarian diet for other reasons, such as a belief in animal rights, concerns about the environmental impacts of animal farming, or religious reasons.
Some people, on the other hand, need specific diets to deal with issues like diabetes (to maintain blood sugar consistency) and coeliac disease (to prevent horrible inflammation or rapid weight loss). Many people who self-diagnose as "gluten intolerant" aren't, because "gluten intolerance" may not even exist as a real thing. The best thing to do if you think you are "gluten intolerant" is, firstly, get tested for coeliac disease, and if it turns out that you don't have that, try a low-FODMAP diet (which is recommended by the FODMAP diet creators to be done under medical supervision, because it's extremely restrictive).
 Fad diets
Fad. As in it doesn't work. Seriously, don't try any of this nonsense.
 The difference?
The simplest solution is often the best. While an actual diet can have many benefits, the fads can often be dangerous or lethal. If you want to lose some weight or build some muscle, reduce some intake and run or walk to the gym. Real easy.