The trysail is a very small sail set in place of the main for storm conditions. It is considered one of the two sails that are never used on a sailboat, the other being the spinnaker. This is probably because people like to turn into powerboats during a storm by dropping all the sails and cranking up the engine.
The problem with this is the extreme motion of the boat during a storm is sediment in the fuel tank can become suspended in the fuel and clog the fuel filter. This is why you always hear of sailors complaining that their engine died at the worst time possible during a storm. If you were relying on your engine, then now you are adrift! Laying a hull under bare poles has been proven a horrible storm tactic, leading the boat to lay beam to the seas and usually capsizing or rolling over completely. When your engine dies in the peak of the storm is not the ideal time to raise your trysail, but at that point, you might be willing to "try" anything, even that "sail".
Being how we don't have the luxury of motoring during a storm (our electric motor doesn't have enough charge to power us through a long storm), we rely on our sails and have actually used the trysail a few times! As soon as we see a severe storm on the horizon heading our way, we rig our trysail. This lets us get everything ready and rigged in calm weather so we are ready for the storm as soon as it hits.
The trysail runs up its own mast track next to the mains track. This allows the trysail to be raised without removing the mainsail.
The sheets of the trysail are run to a turning block aft, and then forward to a large sheet winch. A simple bowline tied to the clew will hold very well during the harshest of blows.
It is important to make sure the lazy sheet is run over the boom and not under it. When it is led over the boom, the sail is able to switch to the other side without issue. If the sheet is led under the boom, the boom will interfere with the sails ability to be on the opposite tack.
The trysail has fallen out of favor by most sailors who state that they could more easily tuck in a third or fourth reef into their mainsail than to rig up a trysail. This is a true statement, trysails take some time to rig in fair weather (before the storm) and could be nearly impossible to rig during a strong blow!
Trysails do have their place though, as they are made out of much heavier sail cloth and are purpose built to weather out a storm. This means that your mainsail is not subjected to all the wear and tear of a storm. If your deeply reefed main were to tear or rip during the storm, you would have no mainsail once the storm has passed. If you tear or rip your trysail during a storm, your main is still untouched and ready to hoist once conditions improve!
The other advantage of a trysail in a heavy blow is it takes the boom out of the equation. Jibing the main can be an eventful occurrence in heavy wind. The boat will heel over greatly and the boom will slam to the other side as it falls past the midship point! Accidental jibes due to wind shifts or purely accident are risky and dangerous during high wind situations. The trysail takes the boom out of the equation and lets you jibe it as easily as you jibe your jib! Simply turn the boat, loosen the windward sheet and tension the leeward sheet; it takes all the fear and noise out of storm sailing.
On a final note, the storm trysail has less going on than a reefed main, which translated into less to break during the storm. The sheet is tied to the clew, run to a turning block and set on a strong winch. Any snags or problems are easy to identify and straightforward to correct.