As BoringComputerGuy said, it depends primarily on what you are studying. However, as a general rule of thumb, I would rank them (from highest to lowest) Durham, Bristol, Edinburgh. Bristol and Edinburgh could change places on that list, but Durham is generally considered to be the best of the three.
For many courses, Durham will also be the most demanding to get into, usually wanting AAA. You are more likely to see some Bs on Edinburgh and Bristol's requirements, though Edinburgh has upped its requirements recently.
However, in terms of general reputation, you may like to consider that Bristol and Edinburgh are both prestigious Russell Group universities. Durham is not, though this is only because of its small size.
The differences between them are not colossal, however - any one of the three you mention would be an excellent university to attend and provide ample academic reputation which any half-decent employer will be aware of.