Detox diets take different forms - some restrict you to fruit or vegetable juices, others to just one type of food, others are supplemented with detox pills - but the idea behind them all is the same. Namely, that our bodies develop a buildup of toxins which make us feel unhealthy and drained of energy, and that these toxins need to be removed. By drastically restricting what you eat and drink to a few special foods, detox diets are supposed to successfully remove the toxins.
Detox diets almost always lead to weight loss. (When you only eat celery for two weeks, such an outcome is inevitable). But once you switch back to regular eating, the weight you lost during the diet usually comes straight back on. Nothing about detoxing helps to change the behaviors and habits that cause weight gain in the first place.
As far as feeling healthier goes, if you completely cut out processed foods, alcohol, and added sugars you may be doing your body some good, but there is no scientific proof that detox diets or fasting can rid your body of toxins any more effectively than the usual processes. In fact, very strict detox diets done over longer periods of time can have the opposite effect.
It's important to remember that your body needs a balanced range of nutrients, vitamins, and minerals to function correctly.