According to WebMD here are the differences:
Cage-Free Eggs
Cage-free eggs are eggs from birds that are not raised in cages, but in floor systems usually in an open barn. The hens have bedding material such as pine shavings on the floor, and they are allowed perches and nest boxes to lay their eggs. However, they may still be at close quarters with many other hens -- just not in cages. That depends on the farm.
Free-Range Eggs
Free-range eggs are laid from hens that have the opportunity to go outside. Smaller farms may keep birds outside under a canopy area. They may travel in and out of a barn at free will or spend some portion of their day roaming outdoors.
Organic Eggs
Organic eggs are laid from hens that may be kept in any kind of caging system, but generally are cage free. They eat an organic feed and don’t receive
vaccines or
antibiotics.
In order to qualify for USDA organic certification, the grains used for the hens’ diets must be produced on land that has been free from the use of toxic and persistent chemical pesticides and fertilizers for at least three years.
Genetically engineered crops are not permitted, and hens must be maintained without hormones,
antibiotics, and other intrusive drugs.
Vegetarian Eggs
Vegetarian eggs are laid from hens that are only fed a
vegetarian diet -- free from meat or fish by-products. Hens are kept in cages or indoors and do not peck any grubs or worms.
Pasteurized Eggs
Pasteurized eggs are eggs in their shell that have been put through a pasteurization process where they are heated to 140 degrees Fahrenheit for three and a half minutes. Eggs are not required to be pasteurized.
Pasteurization completely kills bacteria without cooking the egg. The process can also be done for packaged egg whites used in cooking.
Eating pasteurized eggs is recommended for young children, the elderly, and people with weakened immune systems so they can reduce the risk of contracting a salmonella infection.