Summary: you need to be involved in sinking a ship belonging to a different crew/player(/s) or one of the roaming skeleton ships (identifiable by its ragged sails). I earned the achievement using the very vanilla method of shooting it with cannonballs which will be, by far, the most common combat scenario. Other ways of causing damage to sink an enemy ship (however I cannot explicitly confirm these earn the achievement) are:
1. Ramming it with your ship (note this causes damage to your ship too).
2. Detonating (by shooting it) a gunpowder barrel on or below an enemy ship's deck.
3. Swimming underneath an enemy ship with a gunpower barrel and releasing it so it rises like a depth charge and explodes under the hull.
4. Throwing a gunpowder barrel off the back of your ship when being chased by an enemy ship like a mine for the enemy ship to run over.
5. Boarding an enemy ship and steering it onto rocks so it gets damaged.
Most people will probably run across this one sooner or later anyway. The main point I would make is if you're going for this achievement try not to be a -yoink- about it. if you really want to get this one then get a decent galleon crew together (randoms are in my experience unreliable) if you can to give yourself better odds in combat, but it's completely doable in a sloop. I'd always recommend using a closed lobby, reliable people, and party chat rather than game chat. If you only have two people always go sloop instead of galleon: galleons are hard to handle with fewer than three-four. Brigs are also pretty good ships and if you have three people are definitely the one to go for, however, with only two people the easier sailing on a Sloop is better.
You'll generally run into ships most near outposts or active skulls forts (big skull clouds in the sky) - especially if it's an outpost with few other outposts nearby, but you can run into people literally anywhere in the game.
Try not to just grief people near outposts or near cooperative locations (e.g. Merrick's journal locations/skeleton thrones) though because, fun as it may be for five minutes, long-term it just gets annoying for everyone - plus from a purely practical point of view a ship is far more likely to just run if they're headed into an outpost laden with treasure - especially if it's a sloop and you're a galleon. Unless you're the best sailors and gunners on the Sea of Thieves chances are if a ship is running from you and knows what they're doing you're in for a long and tedious chase. It's far easier to sink a ship if they're as up for the fight as you are.
Requirements: as many wooden planks as you can carry - collect from barrels on outposts/islands, and being aware of the wider situation.
If in a 'normal' fight: keeping your ship afloat is the top priority; it's hard to sink a ship if yours is on the bottom of the sea bed. In the heat of combat especially with a sloop it can be very easy to keep one person on the wheel, one on the cannon, and forget to check for damage. One of you (or you can alternate) needs to head below decks every time you get hit (and just to check occasionally) to fix holes with wooden planks and bale out water with the bucket. Your ship does not need to survive the battle, however, you need to keep your ship in the fight long enough to make sure that the enemy ship doesn't repair itself before sinking.
2). Deaths count.
Requirements: Sword/Blunderbus/Pistol (Eye of Reach can work if you're really good with it, but generally not recommended) with ammo, plus bananas/other fruit (or cooked meat/fish for a health boost) to keep yourself alive (weapons can be selected form the weapons crate on your ship, and you can get more ammo for your ranged weapons from the ammo crate nearby (you can also use the enemy's ammo crate if on their ship). Recommended loadout is Pistol/Blunderbus and sword. Bananas/Fruit can be got from almost any outpost and on many islands from barrels..
Especially if you're fighting a sloop if you can kill one of their crew (either via a boarding action, cannonballs, or sniping - or all of them) it will make your chances of sinking the ship a whole lot easier because of point 1 - their remaining crewman will either have to focus on damage control, or fighting back, if the former then you need to make sure your shots count (see point 3), if the latter then you need to make sure your ship doesn't sink too. Their crewmate will spawn back soon so you have to make the most of this opportunity: keep shooting. if your crewmate is aboard the enemy ship then try to keep the remaining enemy crew distracted while their hull fills up with water as it's hard to repair damage or shoot a cannon or bale out water when swordfighting/shooting a blunderbus.
Likewise: if the enemy kill one of you it makes life much harder so, take care of yourself too: if you get hurt by a stray cannonball
3). Accuracy counts.
Requirements: cannonballs - as many as you can carry - collect from barrels on outposts/islands.
You can put 100 cannonballs into another ship and it might well stay afloat if they're all above the waterline and have damage control. Make sure your shots are hitting the enemy below the waterline (especially on a galleon - you can fill the middle deck of a galleon with holes and it won't sink) and make sure you aren't just shooting one spot. if you keep shooting the same spot there's only going to be one hole (and even with the new damage mechanics thta makes life easier for repairing - although if there's only one enemy crewman alive and you can't see him shooting one spot may be a way to kill him, but I wouldn't recommend that): put shots the length of the ship - some near the stern may be easily forgotten especially on a sloop
4). Moving targets are hardest to hit
Although this partly breaches my don't be a -yoink- point above: if you can catch an enemy crew when they're at anchor (and either not aboard or only have a lookout aboard) at an island/outpost it makes life so much easier: if they stay put waiting for their crewmates to come back aboard they'll be pretty much a sitting duck so you can sail around staying out of their cannon arcs and keeping them firmly in yours. If they're completely stationary you might even be able to anchor up yourself to give you an even easier time of aiming.
5). Gunpowder barrels can be useful
Finally: although it's risky - if you keep a gunpowder barrel on your ship (top tip: crow's nest is safest to avoid accidentally blowing your own ship up) you can use it offensively. You can either swim under an enemy ship and use it like a depth charge (warning: since update 1.0.4 you DO take damage from this yourself underwater now), or you can throw it off your ship if an enemy ship is chasing you and use it like a mine - or shoot it if the enemy isn't quite going to hit it. This is risky as if they shoot it while it's still on or near your ship (unless at the top of the crows nest) then it will cause huge damage to the hull of your ship and probably kill at least one of you. A gunpowder barrel is excellent as an alpha strike and I've used it to sink a galleon when sailing a sloop before - if you are able to release it far enough below the ship that you don't get hurt then you can also follow it up by boarding the enemy ship and killing them while they try to attempt damage control (see point 2 above).
If you're got any further suggestions or corrections then feel free to comment below.
Edit 1: added 'requirements' and note about closed lobbies.
Edit 2: added top level summary confirming requirements, other ways of damaging an enemy ship, and notes about cooperative events
Edit 3: added that Skeleton ships count