Vegan vegetarian - no animal products whatsover
Lacto vegetarian - no meat or eggs but will eat dairy products
Ovo vegetarian - will eat no meat or dairy but will eat eggs
Ovo-Lacto vegetarian - will eat no meat but will eat eggs and dairy
The vegetarian society approves free range eggs and products made with free range eggs. Although vegetarians eat dairy products, most will not eat ordinary cheese as it is made with rennet which is an enzyme found in cows stomach (and extracted from calves slaughtered to make veal). There are other products veggies won't eat - anything made with gelatin, apple juice and alcohols which have been clarified with crushed shellfish, anything containing cochineal colouring etc.
There are also a lot of sweets that veggies (or vegans) don't eat because they have animal derivatives in them (and, no, I don't mean things like Animal Crackers). Muslims won't eat certain sweets, either, because the animal derivatives may not be from 'Halal animals'.
Maidup: There are lots of eggs that people eat, aswell as hens', eg duck, ostrich etc