The simple definition of a vegetarian is one who doesn't eat anything with a face; this includes any type of beef, pork, poultry, fish and insects. I looked online before writing this to see if my personal definition matched the majority and it didn't (no surprise there).
The most common definition I found of a straight vegetarian was one who didn't eat meat, eggs and dairy, but still used animal products. My own take on vegetarianism is the simple definition: if it had a face then I don't eat it. However, not eating eggs and dairy is great too and that's still being vegetarian. There are branches of vegetarianism that have popped up over the years from people who pick and choose what meat they don't eat.
Pollo-vegetarians only eat poultry and pescatarians only eat fish. I suppose if you must eat meat then these two groups of meat are the healthiest options, but a lot of these people seem to be under the assumption that chicken and fish don't 'count' as meat. For whatever reason, these groups exist and are under the vegetarian umbrella term. The next two branches cause my own definition to get skewed.
There are ovo-vegetarians and ovo-lacto vegetarians where the former eats eggs but not dairy and the latter eats eggs and dairy. To avoid confusion or potentially scaring new friends I recommend saying you don't eat meat and then let them find out the specifics; it's more fun that way!
A flexitarian is a term I learned from my dad -- it's someone who goes in and out of being vegetarian. I like extremes so I could never understand why someone couldn't just go vegetarian or stay a meat-eater, but I've learned it depends on the person's situation. If you are traveling a lot for work then it's really hard to impress clients by taking them to a veggie place where steaks are not an option. However, when you are home it's a vegetarian free-for-all so if you can be vegetarian half the time that's better than none at all!
The ultimate level of animal liberation/healthy eating is being a vegan. Vegans don't eat or use anything derived from an animal including: honey, leather, eggs, dairy, silk and fur. If you can be a vegan in this society then you can do anything.
These are the bare basic types to help you familiarize with the current terminology being flung around. I hope this guide helps and happy eating!