There are many different modes, so I'll discuss the differences between them.
Beginner, Easy, Normal, Advanced, Hard and Expert are your usual modes. They have a set number of levels that you must complete. All of these 6 modes make up the Classic level pack. There are the Classic, Swap, Monkey and Extreme level packs.
There are also 'compilation modes', which take you through all of the levels in a level pack, each with different challenges.
For instance, Ultimate Classic takes you through Beginner, Easy, Normal, Advanced, Hard and Expert all in one sitting. ULTIMATE is considered the standard compilation type, just asking you to beat the level pack with a set number of lives. Ultimate Classic takes you through all of the levels in the Classic Level Pack, Ultimate Swap takes you through all of the levels in the Swap Level Pack, etc.
The next is DEADMAN, which gives you only 1 life to beat the level pack. Like in regular gameplay, to get more lives, you must grab all of the diamonds in a single level and reach the goal, rewarding you with an extra life. ...Or if you're absolutely perfect, you could simply beat the mode deathless. That's probably not going to happen...
Next is PANIC, which gives you 90 seconds to beat the mode. For every diamond you grab, 7 seconds get added to the clock. Grabbing a diamond, dying, and then grabbing the same diamond again will NOT give you another 7 seconds, and the diamond will be greyed out to show that it has already previously been collected. You must strike a balance between going fast and collecting diamonds, otherwise you'll run out of time.
Lastly is PERIL, arguably the most difficult mode. This is the same as PANIC, and actually gives you even more time back for every unique diamond collected at 15s instead of 7s. The catch is that you can only store a maximum of 60s on the clock, meaning that you're never far away from running out of time.
The last thing to discuss is the Marathon modes. This takes you through all 500 stages in one huge run, first going through the 210 stages of Classic, the 60 of Swap, the 210 of Monkey, and lastly the 20 of Extreme.
All 500 stages have ILs for Any% and All-Diamonds. If you submit to them, remember to submit your remaining time (IGT), as the charts sort by ascending time.