Vegetarian
Vegetarians are defined in most circles as individuals who do not eat animal flesh and sometimes extends to include any by-products of animal slaughter. This can include all beef, chicken, pork, game, other land animals, shellfish, fish and insects. There are some vegetarians who do eat fish and they are referred to as pescatarians. There are some that eat eggs (ovo-vegetarians) or dairy (lacto-vegetarians).
Vegan
Vegans can be generally defined as individuals who do not eat animal flesh or any animal product. In addition to the no animal flesh, this includes dairy, eggs and sometimes honey.
Plant-Based
The vegan diet is probably the most closely aligned with the plant-based lifestyle with a couple of caveats. Plant-based diets do not include animal flesh of any living animal or by-products of any kind in addition to no dairy or eggs. It is a whole-foods based diet and as a result, it also includes the absence of all processed foods and oil.
There are some minimally processed foods such as whole grain breads, pastas and flours, nut butters, non dairy milks and yogurts, jams and jellies, sweeteners and some condiments that are allowed but in small amounts. The important point to make is even though these foods are allowed, the ingredients in these products is of utmost importance. Consuming foods that have a minimum of ingredients is a good rule of thumb. The closer they are to the original whole food the better.