When food is “processed” it generally means that it is altered in someway in between its source and when you consume it. Many foods we eat today are “processed”.
Processed foods often contain excessive concentrations of salt and/or fat and/or sugar. These ingredients are added to mask the effects of the preservatives/texturizers/emulsifiers/dyes, and to “add back” flavor that was lost in the processing. Each of these ingredients (in excess) can have detrimental physical effects, but consuming them in unnatural proportions/quantities can throw off your body’s ability to gauge its energy/food levels, which can lead to overeating.
Processing often removes elements of a food that are otherwise beneficial- e.g. the skin of an apple contains lots of soluble fiber, phytonutrients, and other good stuff, but gets stripped away and discarded for most processed foods.
Many chemicals can be introduced in food processing to add flavor, texture, color, aroma, shelf life, etc., and these can often have real (or imagined) side effects. Olestra, for example, can give you diarrhea, and make it difficult for your body to absorb other nutrients. Nitrites, used in cured meats (the original processed food) are associated with some carcinogens.
Juice is processed by taking fruits and removing the fiber (solid bits). The fiber in fruit is like a natural constraint. eg. it would be tough to eat 10 apples, but easy to drink 10 apples worth of juice. The juice is calorie dense compared to the fruit making it easy to over consume and gain weight. Milk and juices are often pasteurized, pasteurization, the heating of liquids to kill bacteria, being the process they were subjected to. So its tough to generalize and say all processed food its bad for you. It really depends on the process its self.
High fructose corn syrup is the result of processed corn products (glucose) that result in the sweet maple like syrup that they put in everything (fructose). Corn syrup isn’t necessarily bad for you in small quantities, but it is very calorie dense. So eating large quantities will result in weight gain.
White bread is also a processed food. They process the wheat by removing the tough outer shell of the wheat where all the nutrients and fiber are found. You end up getting all the calories from carbohydrates without any of the fiber or nutrients, and often people refer to this as “empty calories”.