Dungeons have Normal and Hard. Well, some of them have Hard; and those are essentially "return to the dungeon with brand new mobs, layout, bosses, etc.", so they're almost entirely different experiences; they're no harder or easier than other dungeons with similar requirements.
Trials have three settings, but only a couple matter. Story/Normal and Hard can be considered the same classification, actually, except for the first three trials. This is because after those first three trials, all trials either don't list a difficulty, or their default difficult is listed as Hard. Basically, everything from 50 onwards defaults to "hard", even if it's part of the story; and just means it's 8-man. For the three trials that do have both a Story and Hard separately, the Hard variants are more complicated and involved fights... but they also allow significant overgearing, and may actually be easier (especially The Howling Eye). Trials in this classification generally only reward crafting materials or triple triad cards, though a couple drop weapons.
The designation for trials that does make a difference is Extreme, often shortened to EX. All Extreme trials are completely optional fights only accessible after clearing the Hard modes, and they're much more challenging than their default versions. Some are easier than others to be sure, but these aren't fights you'll do well in trying to go in blind, unlike other trials. They're generally either tuned to be very punishing, or have mechanics that require coordination or skills not tested elsewhere, and just generally demand a lot more out of their players. That said, they're not out of reach of the average player to clear. They reward completely different items than the Hard mode versions do, including unique accessories, weapons, mounts, and materials.
Savage is a designation specific to raid content. It cannot be entered by queueing solo through the Duty Finder, unlike the standard raid difficulty level; you have to preform your party for it. There's currently two Savage raid sets. One for the Second Coil of Bahamut raid series from ARR, which rewards no unique gear but does reward unique achievements and titles on clear. The other is for the most current raid, Alexander; where Savage mode drops higher level versions of the normal difficulty raid gear which can be dyed. Much like Extreme, these demand vastly higher performance from all players present, but even moreso; they're designed for a very tiny amount of the playerbase to be able to clear.