The way I've always heard it called, based on the orientation of that picture, is that "height" is up/down. The card slot bracket is where 'full height' or 'half height' come from. Some low-profile PCs need a 'half height' card that has a card slot bracket that is half the height. (For ISA/EISA/PCI/AGP/PCI-e, "full height" is 107 mm tall, and half height (properly titled "low profile" in the PCI spec) is 64.11 mm tall.
"length" or "depth" refer to the distance from the card slot bracket to the other end of the card. (Left/right in this picture.) The PCI specification officially allows cards up to 312 mm long, but good luck finding all but a massive server chassis that support that. (Although the just-released Radeon HD 5970 comes darned close at 309 mm long, including shroud.)
Finally, "width" is used to refer to how many card slots worth of space it takes up. The vast majority of cards are one-slot wide; but most high-end graphics cards nowadays are two-slots wide.
These three definitions are leftovers from the days when nearly all PCs were desktop models, with the motherboard mounted parallel to the floor, and the cards sticking perpendicularly up from the motherboard. When doing work on a modern tower PC, this still makes sense, since most technicians put the tower on its side, so the motherboard is again parallel to the floor, cards sticking up. (I saw an argument on another messageboard recently that quickly turned into a flamewar, where one guy was just ADAMANT that "half-width" meant what the PCI org calls "low-profile" (what I call "half height",) and "double height" meant a video card that takes up two slots (what I call "two-slot width".) I chose not to get involved in that flamewar.)
So since I wandered so far, I guess I'll answer the unasked question: This card is a half-length card, not a half-height card.