It is very difficult to avoid animals products in this 'modern day and age'. Here is a list of some common things that surprisingly contain animal derivatives and others that are safe.
CALCIUM PHOSPHATE: This is often used in English muffins, breads, cereals, and other baked goods as a preservative. This one is really just a fancy word for ground up animal bones and teeth.
CASEIN: This is a product made when milk is heated with an acid, like lactic acid. This stuff mostly occurs in "no-lactose" soy cheeses like Soyco or Soy Kaas. The labels say "lactose-free" (lactose is another milk derivative), but that doesn't mean they are therefore vegan, as we used to incorrectly assume. Soymage soy cheese is 100% vegan, but it's kind of gross. Casein is also used in plastics, adhesives, and paint manufacturing.
CASEINATE: Casein mixed with a metal, like calcium caseinate or sodium caseinate.
CHEWING GUM: Some chewing gums contain glycerine. Wrigleys gum contains a vegetarian source of glycerine.
ENVELOPES: Apparently most envelopes have a synthetic glue on them, not an animal or fish based glue.
MARGARINE'S: Can contain fish and other marine oils. Many margarine's contain whey.
MOHAIR: From goats. They can be sheared or skinned.
NOUGAT: Usually contains gelatin.
ORGANIC: Dried blood, bone/hoof meal and fish meal can all be used a fertilizers etc. Try finding out about Veganic Gardening as an alternative, by using seaweed fertilizers which are widely available.
PASTA: May contain egg, especially if fresh. Some pasta in Italy contains squid's ink; this can easily be recognized because the pasta is black.
PASTES: Glues. May be animal or fish derived.
PASTRY: Animal fats used in most shop-baked pies etc. Check ingredients.
PHOSTATES: Derived from glycerol and fatty acids. May be from animal bones too.
PHOTOS: Developing paper contains gelatin.
POSTAGE STAMPS: These do not contain an animal or fish glue.
PROGESTERONE: A hormone. May have been taken from the urine of a pregnant mare, and could be used in hormone creams, etc.
RENNET: An enzyme taken from the stomach of a newly killed calf. Used in the cheese making process. Look for rennin or the words "made without animal rennet".
RUBBER: Processed with animal products.
SHORTENING: Can be made from animal fats. Used in the food industry especially pastries and biscuits.
SOAP: Most soaps are not vegetarian because of the tallow (animal fats) and because many will have been tested on animals.
STEARATE: This usually comes in the form of calcium stearate, and it is found in hard candies like Gobstoppers and Sweetarts as well as other places. It comes from stearic acid, which usually is derived from tallow, or animal fat. Stearate is also used in vinyl's (like car seats) and plastics.
SUEDE: Leather.
SWEETS: Watch out for gelatin, e.g.: wine gums. Nearly all mints e.g.: Polo, Trebor, Extra Strong etc contain gelatin. See also Nougat.
VIOLINS: Traditionally violins are stuck together with an animal based glue. The bows are usually made from horse hair.
WHEY: Liquid part of milk.