Usually tris, someone my answer that question thinking in quads but nowadays when someone ask "what's the poly count of this mesh ?" he usually means "what's the tris count ?".
In any case it is better to specify, so instead of asking "what's the poly count" it's better to ask "what's the tris count". Same thing for answering.
It's actually better to think in vertexes. Vertex count gives a more accurate reflection of how big the asset is going to be in memory and how long it will take to draw.
You also got to keep in mind, you are talking about one small part that works in a much bigger whole. Don't get so focused on the number of vertices of your model that you forget about all the other factors that will effect performance. There's a lot, and in the beginning it can seem overwhelming. But the best thing to do is stress test your work in a game engine. Make a small demo game, drop a lot of your work in there, learn how to use the profiler, and then you can experiment with things like LOD's and texture atlas and different shaders and see what that ends up meaning for the end user.
This way you get directs eyes-on experience which will quickly make you understand theories more intuitively.
Personally I was surprised when I made my first game just how many millions of triangles even low-end computers could handle. Most of the optimization work I ended up doing was in-engine : lighting and post-process stuff.
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In any case it is better to specify, so instead of asking "what's the poly count" it's better to ask "what's the tris count". Same thing for answering.
http://wiki.polycount.com/wiki/Polygon_Count#Triangle_Count_vs._Vertex_Count