Term for easing up sails in a heavy storm
Solution 1:
There isn't a single way to reduce "sail power", hence no single term for this. There are several terms for specific actions that may or may not do the job, depending on the circumstances being reacted to. Reviewing some of the more likely and less likely ways to accomplish a reduction of sailing power, you can see that there's no word that is exactly what you're looking for:
One may douse the sails, if the rigging is such that suddenly dropping the sails is possible.
You can ease (not "ease up") anything under tension, including sails' rigging. The lines controlling the sails are the sheets and the most likely candidate for emergency easing. This can be done quickly, though loose, flapping sails can be dangerous, especially with larger sails, so easing an entire sail is not a default emergency measure.
The boat can be put in irons (i.e., robbing it of all wind power and getting it "stuck") by turning it directly into the wind, but that's a fairly drastic effect on one's control of the boat for little gain. Conceivably, in an emergency that could be an intermediate action to gain some breathing space to deal with the sails directly.
More abstractly, you can bleed wind from the sails, but that doesn't describe a specific action. Actually, this may be closest to what you're looking for, as it describes a specific kind of reaction to an emergency. It's not a general term, though, as it doesn't cover dousing or reefing, for example. It's an appropriate measure when just a little bit less wind will correct a problem or impending capsising, so possibly not the term to use when responding to a sudden storm.
For reference, reefing sails actually means to tighten them to their minimal area without entirely removing them as wind surfaces. Furling is to wrap a sail so it has no wind surface. Depending on the rigging, reefing may be a necessary part of the process of furling a sail.
Solution 2:
Generally, any action to reduce the wind surface of the ship's sails or their ability to pull on the ship will reduce the "power" of those sails. There isn't one way to do it, and so there isn't one term for it. The main methods on a square-rigger (which includes most of the iconic "tall ships") are to reduce or minimize the sail's surface area, or to turn the ship so that the wind blows across them instead into them.
The normal way to stop a ship in open water usually doesn't involve taking in sail (at least not till after the ship has stopped); instead, the sails and the ship are positioned so that all the forces on the ship are cancelling each other out. This process is called "heaving to", and on a square-rigger it's done by turning aside to the wind, then literally heaving the jib sails to windward (pulling the windward sheetline taut and letting go the leeward); this angles them so they push back against the rest of the sails, bringing the ship to a stop; the next command, if they are to remain stopped, is usually to "take in sail" or "dowse canvas".
The fastest way to properly stop using a sail is to "take it in" or "put it in its gear". The sheetlines (and tacklines of a course sail) are loosened ("let go"), slackening or freeing the lower edge of the sail, and then the clewlines and buntlines (stretching from the yard down the height of the sail to the lower edge) are hauled from the deck to pull the sail up to the yard. Originally, the buntlines were simply attached to the bottom of each sail, and as they were hauled the sail folded in half. That advanced to a system where the buntline goes around the entire sail and is anchored on the yard; as it's hauled the sail is pulled more compactly up under the yard. The sail is then said to be "in its gear". Only the course (lowest) sail of each mast could be put in gear completely from the deck; the sheetlines of the topsails, topgallants and royals were usually attached to the yard of the sail beneath, and so to furl those the sheets first had to be disconnected, by climbing out to the yardarm of the next lower sail and freeing the sheetline. Not a job for the faint-hearted (really, there's no job aboard ship that's for the fainthearted).
If there's time and ability, and the sail isn't to be used for a while, the crew would then properly "stow" or "furl" it by tightly binding it to the yard with its gaskets. This is normally done prior to a storm if at all possible; if the ship is caught in a sudden squall, putting the sails in their gear is usually enough.
To lower the center of gravity of the ship (useful in a storm), the halyard which supports the yard spar of the sail can also be let go, which on older large ships allows the yard to be "sent down" all the way to the deck (an operation requiring a lot of manpower to be done quickly and not usually done while actually in a storm). On newer and smaller ships this usually doesn't happen, but those ships often have "lifting yards" which can be raised and lowered along a section of the mast; in their lowered position they're easier to get to and the weight of the yard and sail is carried lower to reduce rolling in high seas.
As a stopgap in a hard blow, the ship may turn 90* to the wind and "square sails"; the "braces" (lines controlling the angle of the yard to the ship) are hauled to literally square the yards, and thus the sails, perpendicular to the ship's center line. That would cause the wind to blow across the sails instead of filling them, giving the crew time to strike and stow them. This is useful in high straightline winds, less so in cyclonic winds (where the direction of gusts may change often).
Completely freeing a sail by cutting it loose was an absolute last resort, usually when one or more of its brace, tack or sheetlines had failed, causing the crew to lose control of the sail, and the sail was threatening to take its yard and maybe the mast with it. Freeing a square-rigged sail requires climbing up onto this unstable sail's yardarm and cutting all its robands off the jackstay, freeing the mast from the yard, then cutting all the clewlines, buntlines, tacks and sheets to let the sail blow away. The sail was very unlikely to ever be seen again, much less recovered, and there was always a chance that the sail could foul on one or more other lines as it blew away, creating even more problems. Most often, this would be done to the jibs; on a tall ship these were typically the easiest to get rid of in this manner, the least likely to foul, and the least vital sails when underway (they help trim the ship and are used to heave to but none of these maneuvers absolutely require them, and they're not used at all when sailing before the wind).
Understand that even in a storm, some sail has to be hung if the ship isn't anchored; a ship with all sails furled can't control where it's blown, and just tosses about in the waves. So, you have to hang some canvas; good options are the course and lower topsails of the foremast. These normally can't be reefed on a square-rigger, but that's why you have so many different sails, so you can adjust how much wind you catch and how much is aloft.