Fish is DEFINITELY meat. Fish also feel pain. I wrote about
why eating seafood should be avoided in another thread.
Vegetarian -- no meat at all. No food that caused an animal to die. No fish, no chicken, turkey or other poultry, no red meat, no pork. No other meat -- cuy (that's guinea pig), rabbit, duck, deer, elk, etc. etc. etc.
No broth or stock made from meat (chicken stock, beef broth, bonito [dried fish], etc.).
Also vegetarians try to avoid all other products that caused animals to die. No gelatin(e) (made from bones and tissues, found in Jell-O and marshmallows), no lard/suet/tallow (animal fat), no cochineal or carmine (made from crushed insects), no rennet/pepsin (found in many cheeses, made from calf or pig or other animal's stomach, usually not listed as an ingredient, or listed as "enzymes"; plant-based enzymes are not distinguishable in the ingredients list so you must call the company and ask which they use), aspic (a kind of gelatin like product made from meat), no fur, no leather, no down (feathers pulled off usually dead birds), no caviar/roe (fish eggs, but unlike chickens, the fish is killed to harvest them), no chitin (from crab shells), no shellac (from bugs), no isinglass (made from fish swim bladders, in some beers and wines), no keratin (from animal hooves, horns and hair), no fish scales (found in some cosmetics), no musk (from the glands of deer, etc.). Vegetarians may avoid regular silk (worms are boiled alive to harvest it) but may buy "peace silk" (made without killing the worms).
Vegetarians generally do eat eggs, milk, and honey. Vegetarian generally means "lacto-ovo vegetarian". Some may give up one or another. Lacto-vegetarians eat dairy products but not eggs. Ovo-vegetarians eat eggs but not dairy.
Vegan -- No animal products at all. None of the things vegetarians avoid, and more. No meat, no eggs, no dairy.
In addition to the things vegetarians avoid, vegans must also avoid products like whey, lactose and casein (all made from milk), honey and beeswax and royal jelly (from bees), lanolin (from sheep hair). Also no silk, wool, alpaca, angora, mohair, cashmere, etc. They also purchase sugar that wasn't processed with bone char.
These are NOT complete lists.
And, expect to make some mistakes until you get used to reading ingredients. You'll get really fast at it.
Also, no one is perfect. We only aim to be as careful as we can. And it's not about personal purity -- it's about saving lives and eliminating suffering.