This is difficult because:
- If you say you're vegetarian, you may get served fish because many people don't realize it's not vegetarian
- If you say you're vegan, lots of people won't understand and will think it's vegetarian.
- If you say you only eat plants, lots of people will still serve you cheese and eggs
- It you try to just enumerate all of the things you can't eat, you may skip some culturally-relevant items, (eg crema in Mexico, paneer in India, leverpastej in Sweden, or furikake with bonito flakes in Japan)
Google Translate
The best thing to do is probably to just print a sheet of paper explaining veganism in the language of the country you're visiting before you go. Something like:
Hello, I am vegan. I am allergic to food from animals. I can only eat plants.
I cannot eat: meat, fish, eggs, cheese, ...
I can eat: rice, beans, vegetables, fruits, mushrooms, nuts, seeds, grains, etc
It would be a good idea to just spend a few hours googling "vegan in country X" before you do this.
After reading from other people's experiences, try to figure out what are the most common non-vegan ingredients (eg milk derivatives, fish condiments, etc) served in that country and add them to the list above before you translate it.
Vegan Passport
The Vegan Society maintains a little book called the Vegan Passport that is designed for this purpose.

I haven't owned one, but my understanding is that they translate what veganism is in every language specifically for this purpose: so you can hand it to your waitress/waiter and they'll know what you can and cannot eat.
As far as I know, they take into account culture here, so they will specifically point out commonly-used ingredients for the given country that you cannot eat.