Depends on which country you are in, but in the US almost every processed food contains an animal product, such as coloring from crushed insects (carmine, aka cochineal).
Cochineal-based dyes and derivatives (Blue 2, also known as Indigo Carmine [sometimes used in nylon sutures]), E120, carmine, caramine, carminic acid, crimson lake, Ponceau, E124, Cochineal Red A) cause allergic reactions in no more than 1 in 10,000 consumers. The Center for Science in the Public Interest has called on the FDA to remove insect-based dyes from the approved list of dyes.
- http://flaggedrevs.labs.wikimedia.org/wiki/Cochineal
Always read labels and if there is anything listed which you cannot identify as a non animal food, don't eat it. You can contact the company to ask whether the ingredient is from plant or animal source if you really want to know.
Gelatin can be an ingredient in jams, jellies, many candies, yogurt, Jello, gummy bears, gummy worms, all the gummy candies.
Be aware that most white sugar is filtered through bone char, so unless it's labeled vegetarian, vegan, organic or raw sugar, any item which lists sugar as an ingredient most likely contains sugar which has been filtered through animal bones. Many alcoholic beverages are also filtered through animal bones. One type of beer even throws a whole chicken into the vat during the beer making process.
Many vitamins are from animal sources. Vitamin D3 is from animals (D2 is not). Calcium is often animal sourced. Omegas are often animal sourced (omega 3 is in flaxseeds, walnuts). Every nutrient found in animals is available from plants and/or non animal sources. Every nutrient.
Animals are in so many foods, it's easier to list the foods which do not contain animals. That would be fruit, vegetables, whole grains, legumes (beans, peas, lentils), nuts and seeds.