What is a "red brick"?
One of 6 particular universities.
A similar term in the USA would be something like Ivy League:
It's a phrase that someone from the associated region of the world can generally go "Oh, you attended a <terminology> school?" and immediately have certain stereotypes and assumptions (accurate or not) come to mind.
But it's quite strong for CS/AI, as they have the largest CS department in Europe, producing the largest amount of world-leading research (according to the latest government-organised Research Assessment Exercise).
For undergrads that means there is an unparalleled variety of courses you can take, both in depth and breadth. I remember I had a choice of about 25 courses in my 3rd year and over 50 in my final year. For most British universities, your choice is rather limited - usually between 8-12 courses each year. Another bonus is that you get to work with leading researchers on your final year project. E.g. some of my friends were supervised by Phil Wadler - the guy was one of the principal inventors of Haskell and, with Gilad Bracha and Martin Odersky, designed Java generics and the extensions of the Collections framework.
So, Edinburgh is quite special and CS/AI is a particular focus at the University, so it's a shame not enough people know it.