Sorry not familiar with Hartpury specifically but the basic levels are:
-Intro - walk and trot only, simple school movements
-Prelim - walk trot and canter, simple school movements
-Novice - medium trot and canter, counter canter introduced, slightly harder movements
-Elementary - basic lateral work, collected trot and canter, simple changes of canter lead (through trot and walk)
-Medium - harder movements e.g. walk pirouettes, half pass, and extended trot and canter
And then it goes advanced medium, prix st georges, inter 1 and 2 and finally grand prix which is the highest level and what you see at the Olympics etc., includes the hardest moves such as piaffe and passage - but even Olympic riders will still compete at the lower levels on their less experienced/younger horses. There are a whole variety of different tests at each level, find them by googling e.g. Prelim 7, Novice 12 etc., you usually have to pay for the sheets which contain the mark schemes etc - but there will usually be multiple you tube videos of people riding the tests, some with commentary explaining what the judge is looking for so that's a good way to learn. Basically at all levels it's about the scales of training, the horse working correctly and in a relaxed fashion, performing the right moves as required by the test accurately, the rider being quiet and effective.
Different affiliated and unaffiliated competitions will have different structures, some will just simply run multiple intro, prelim and novice classes on the day (with a different test for each) and everyone competes against everyone else, but quite often classes will be divided into 'open' and 'restricted' (for less experienced combinations) sections, or there may be senior and junior classes (based on age of rider), young horse classes (for horses aged 4, 5 or 6), there are qualifiers for regional and national championships, specific classes, leagues or championships for horses of different breeds (e.g. native ponies or retrained racehorses) plus a lot of riding clubs or venues run summer or winter leagues so you may earn points towards those...
Basically it can be confusing to know what to enter, but pretty much any rider with a relatively well schooled and quiet horse should be able to go out and do an unaffliated intro or prelim test so no need to boggle your mind too much if you want to give it a go - find your local venue and enter whatever classes you're eligible for, buy the test sheets and practice them in advance then bling yourself and pony up on the day and give it a crack! You'll always get a judges comments sheet afterwards with (hopefully!) constructive feedback on where you can improve!