As mentioned, there are now light and heavy aquifers. The heavy ones require the usage of one of the "traditional" aquifer breaching methods (there's also a new "chicken run" version, but, regardless, you need specific methods to deal with it).
Your aquifer is heavy (where you dig) if it fills up quickly. If the dorf can be ordered to dig down another level after the "wet stone" cancellation and succeeds in doing that, it's a light aquifer, and if the dorf fails it's a heavy one (or an exceptionally slow dorf...).
Getting through a light aquifer is fairly easy (but it's still possible to screw up: I've done it, and I'm sure others have as well): Dig down to at least two levels below the last aquifer level (the level just below has water raining from the ceiling, which obviously is no good). Once there, dig out some area, with 10*10 tiles being more than enough, for water to flow out into and evaporate.
Initially, my staircase is a single tile wide. Once I've dug out an evaporation area at the bottom I can then expand and seal it. My preference is a 3*3 staircase with a pillar at the center tile, but you can use other shapes (using ramps is a completely different issue, as you get leaks from the ceiling: making a caravan path down requires a fair bit more work, but it's not impossible, even with a heavy aquifer). I expand my staircase from the top down, one level at a time. This means that I dig out the remaining staircases on the current level plus all the tiles directly adjacent to the staircase, except the corners (as mentioned, leakage doesn't happen diagonally), and, in my case, the center tile (or it will leak), and then build walls on all of the tiles dug out (again including the center tile). You'll get an occasional cancellation because of water on the tiles, in which case you have to resume the suspended construction.
In my case, I let my two miners do all of the work, with them having no tasks beyond digging and construction, and nobody else constructing (the miners are then always close to the construction site and will switch to construction once digging is done, rather than having other dorfs that may be half across the map taking up the construction tasks, but you may want to organize it differently).
Usually an aquifer extends only through one or two layers of soil. If, however, it extends down into the rock (e.g. into sandstone or conglomerate), you switch the "sealing" method to keep the water out from construction to smoothing, so instead of digging out the adjacent tiles you order them smoothed (make sure to enable stone detailing on dorfs [in my case the miners]) as smoothing seals the rock so it won't leak.
If you dig out the staircase all at once you'll get more water flowing down, although that's usually not a problem if you're not dragging your feet in the digging (deciding you need to dig out something else halfway through is a bad idea: once started you need to get it done). If you try to dig one level at a time and seal it as you go you'll probably find that water slowly accumulates such that you're eventually blocked from continuing digging.
In short: the key to dealing with light aquifers is to dig through it and then dig out space below for the water to evaporate.
It is possible to disable aquifers. The LNP provides an option to do so, but as far as I understand it involves changing the raws to have none of the inorganic materials support aquifers, rather than just change an init file setting.