Button down

Neither of those is a natural expression.

Consider these.
a button-down collar is a collar with buttons that hold down the collar points.
You can easily google a picture.

button up is a phrasal verb meaning 'fasten all the buttons'

Thank you Clive. I'm confused, though.
In the novel I'm reading, the author (American) describes someone as wearing slacks and a button down, which I assumed was short for button-down shirt.
If I google "button-down shirt" or "button-up shirt" I get images of pretty much the same thing–a shirt with buttons running the lenght of the front. That's why I wondered if there was a difference.

I would understand a "button-up shirt" to be a shirt which fastens with buttons down the front -- what in the UK we would usually simply call a "shirt" -- and a "button-down shirt" to be one with a button-down collar. This is also what it says at http://en.wikipedia.org/wiki/Dress_shirt

Thank you GPY.
I had started reading the wikipedia page that you also reference, but I stopped before I got to the part where it would have actually answered my question.