Picric acid is used in qualitative organic analysis labs. As AWK pointed out, a picrate is a salt of picric acid. Picric acid itself is a high explosive, having explosive performance similar to (slightly better than) TNT (trinitrotoluene). Many of the metal salts are highly dangerous explosives, as they are friction, impact, and heat sensitive. I would be careful if you have to handle it. Picric acid itself is somewhat friction sensitive, so it's almost always handled in a water solution.
Anyway, the picrate part means it has the 2,4,6-trinitrophenoxy anion. You can have whatever as the cation, and it gives it the first part of the name. In your case, you'd have triethylammonium picrate.
Just as a side note, ammonium picrate is used in making whistling fireworks. I always thought that was pretty cool.