What are the different dungeon modes?

Solution 1:

The story mode is actually a side story, an add-on to the personal story. Explorable mode is usually providing harder quests in the same dungeon.

Most dungeons have 1 path in story mode, and 3 different paths in explore mode. There are exceptions to this rule, however.

• Twilight Arbor doesn't have 3 clear paths since the beginning to choose from, but has in fact 1 or 2 intersections. The possible paths still sum up to 3, though.
• The Ruined city of Arah has 4 explore paths, instead of 3.
• The Fractals of the Mists is particular, and is actually composed of 9 mini-dungeons. A run through the Fractals will pick a combination of 3 of those mini-dungeons, to which will be added a fourth one (fixed one), if the difficulty scale is an even number.

So if we sum it up, it makes seven "regular dungeons" with four paths (1 story + 3 explore) and one with five paths:

7 x (1+3) + (1+4) = 33 different experiences from the regular dungeons.

The fractals can be counted on several ways.
Technically, there are only 9 different zones. However, if you count each possible combination as a different instance, the total goes up to:

  9*8*7 (for uneven numbers)
+ 8*7*6 (for even numbers)
=   840 different dungeons

If you want to be particularly specific on anything which defines a "different" experience, you could account for the difficulty level (80 of them), as well as the order in which the combinations come (and multiply by 6), which would leave you with a total of...

403200 "different" fractal experiences.

But that would be slightly exaggerating.