In today's t-shirt culture, I've frequently seen dress shirts without button-down collars erroneously referred to as "button-down shirts," presumably because they have a placket with buttons down the front, and one usually buttons them from top-to-bottom.
But those aren't necessarily button-down shirts; as you probably surmised from the previous paragraph,"button-down" shirts are shirts with buttons on the ends of the collars. They supposedly originated around the turn of the 20th century with English polo players, who devised the style to keep the collars of their shirts from blowing up into their faces while they were playing. Brooks Brothers quickly imported the style to the United States, where it quickly became a classic.
Shirts with a separated front that encloses entirely with buttons (as opposed to a pull-over, which may or may not also have buttons near the top) are button-up shirts.
Even so, the term is less than ideal, because not all button-up shirts are dress shirts.
I prefer the following, more precise terminology:
Dress shirt (should need no explanation, but the men I see wearing dark shirts with ties indicate that I'm wrong; see my post on dress shirt colors, and maybe we'll discuss dress shirts at length someday)
Sport shirt (a casual shirt never to be worn with suits or blazers, and generally not even with very casual odd jackets; these shirts often have two chest pockets with button flaps, and are usually in a bold pattern like plaid; they're also often in a non-dress shirt finish, like flannel)
Button-down dress or sport shirt (a shirt with buttons that fasten the tips of the collar points to the body of the shirt; with dress shirts, this style is the most casual)
Understand the difference between a button-down shirt and a button-up shirt, then try to strike the vague "button-up" term from your vocabulary.