Thanks so much for the help but I honestly have no clue what you are talking about. I'll try to get some measurements sometimes and maybe that'll help.
Okay, here's as simple as I can make it. The term "half height" does not have a set definition or standard, as best I can tell. Any short card can be called half height, regardless of how it measures in other dimensions.
The term "low profile" is defined by the group that makes the standards for PCI connectors. There are certain measurements that have to be met to be considered "low profile".
I'm not sure how best to describe what is height, length and width without an illustration, so I'll try and skip that for now.
The mounting bracket is the angled piece of metal used to screw the card to the back of the case. Mounting brackets can be of two heights, and some are double-wide. A double-wide bracket will cover the opening for the slot next to it, and won't fit in cases that have that slot full with an existing card, or where there is no space for another slot at all.
If the slot next to your video card is empty, you don't have to worry about how wide card you get. If it's occupied, or if you don't have a slot there at all, you'll have to make sure the bracket is single wide, and the heat-sink on the card isn't too large.
Does that make sense?