There are 5 types of Elite Bejeweled 3 Badges, I will list all of them and how to achieve them.
Pretty simple, all you have to do is go to 'Quest' mode and then from there, complete all 8 modes for that 1 relic. There are 5 relic's overall and you must complete a total of 40 modes, majority being the same, but they get tougher. After you get this, it'll pop up saying you've earnt this badge and most likely the achievement for earning 1 Elite badge.
This one's pretty tricky, but not the hardest, best game mode to do this on would be Diamond Mine, basically what you have to do is match up 2 hypercube gems with each other, so if you're going for this, you must make sure the the hypercubes (when you eventually get 2) are in the same column or in vertical from each other. Once triggered it'll clean the whole entire board of gems up giving you the badge after you finish the game mode.
Hardest one, but can come with luck, best game mode to attempt to get this would be Diamond Mine as well, the reason being Diamond Mine is because you start off with 4 different colours and it makes it that little bit easier to obtain. If you really need to, you can keep pressing the 'back' button to reset the level if you wish. Also a Supernova gem is like a Star gem, but instead it'll take up 3 columns and 3 rows on detonation. Once the game mode is completed, you'll get your badge.
Basic badge, all you have to do is reach level 10 in classic mode, it can be a little tough sometimes trying to make it to the end of a level without ending up with no moves, but I have a tiny tip for survival, that actually helped me get to level 10. What you have to do is keep the bottom half gems a back-up, so you know you've got something to keep you in the game, so what you're basically doing is trying to get rid of everything in the top half before needing to use the bottom gems. If you haven't done quest yet, this will most likely the badge to unlock the 'earn 1 elite badge'. Once you've completed the game mode, the badge will be yours.
Not the hardest, but not the easiest either, you're going to need to play this for a while before you achieve this badge, I currently only have a rating of 14, nearly 15. The best game mode(s) I may suggest to boost this would have to either be Diamond Mine or Lightning Mode. Whlist doing Diamond Mine, you'll also be getting progress towards the Flame gem, Star gem and Hypercube badges. That is all I can really offer towards this badge.
Hopefully this guide was somewhat helpful, this being my first guide, if you vote negative, please leave a comment why and I might be able to try fix up a section, so it looks/reads better. Thank you.