Distinction: Refers to things that stand out, that are different - distinct from - other things around them. In some qualifications in the UK for example you can earn a 'distinction' by scoring over a certain percentage, this makes your qualification 'distinct' from others of the same type that don't have a distinction.
Diversity - refers to variety in any situation where there exists a possibility for it, e.g. a diversity of opinions, a diversity of options, diverse ideas, diversity in demographics and so on. Basically means a lot of different and potentially unlimited possibilities within some area of thought, or some group of people or things.
Variation - limited diversion from the mean, in a set of data, in opinions, in habits etc. E.g. 'there is some variation in the answers given to the questions we sent out, but overall they are quite similar', or 'there is some variation in his daily habits but he tends to follow the same routine pretty much every day'.