Most people connects E numbers with something unnatural and something that does not belong in foods.
But... many of the E numbers are naturally occurring substances and the E numbers system is simply a classification system for food additives. Food additives are added to the food for different reasons ranging from food safety to taste.
Some E numbers should be avoided though, that is the AZO-colours; E102, E110, E122, E123, E124, E129, E151, E155 and E180. Studies have shown that these could have negative effect on human brains, especially with children. These are forbidden in some countries, and the colors are found in different kinds of candies. So check the table of contents of your product before buying it.
By the way, the picture above, showing tons of chemical substances and E numbers? That is simply what a kiwi-fruit contains, if it should have been marked with a table of contents.