I know that minimal light pollution is a must for stargazing, and a place which is away from civilization is better.
Does altitude of a place matter for light pollution?
Does it affect the quality of star gazing?
A good question, and in the early 2000s John Bortle published a categorization of a variety of conditions, with descriptions for each category. It is the commonly used scale to describe to others the sort of conditions at a location.
Probably one of the more significant factors provided by a dark sky site is: how faint do stars have to be for you not to see them anymore (overwhelmed by light pollution). There are specific stellar regions, each a triangle, and the idea is you could how many stars you can see in the triangle, and you look the number up in a table, and it will tell you the magnitude limit you are perceiving at that site.
Yes, high altitudes also help - less air between you and the stars means better seeing conditions/less atmospheric distortion.
Being able to see more stars IS better for star gazing. Living in a city, I am continually frustrated at the poor skies, and love being in dark sky locations like the International Dark Skies Reserve at Tekapo, New Zealand.
The other answers have some good points: Lack of light pollution does make for much better skies, and higher elevation does help by eliminating some of the obscuring atmosphere (of course, you need to be careful how high you go and how quickly, as lack of oxygen can cause problems of several sorts).
I'd like to add a couple things:
Yes, altitude does play a role. The higher you go, less photons from civilization will reach you, as most of it will be scattered by dust in the air.
Also, as to answer "what qualifies as a good place", as if you are asking for a criterion, a popular one is : "If you can see all the stars of Little Dipper, well that's a 'good' place to observe".