Note: Parts of my reply here come from an earlier reply of mine here.

Ships were actually rarely sunk during this era. They floated on their timber and it often took storms breaking them apart for them to sink. They could catch fire and burn to the waterline, but that would still often not sink them.

They would be damaged, the crew would be wounded, maimed or killed, they would be demasted or derigged and be dead in the water, but they would rarely sink.

Note for example how USS Merrimac burned to the waterline and the hull was still usable to construct the ironclad CSS Virginia.

Ship of the line (and modern frigates akin to the USS Constitution) were heavily built during this era - up to a meter of hard oak wood was not uncommon, so the ships had quite a bit of "armour" and could resist fire quite well. The USS Constitution gained the nickname "Old Ironsides" in the battle with the HMS Guerriere, when the British 18pdrs proved incapable of penetrating the hull of the more modern and heavily built US frigate.

During the latter half of the 18th centry, naval combat turned pretty indecisive. Fleets lined up and blasted each other, causing considerable casualties and damaging each other a lot, but seldom actually disabling or destroying their opponent.

For example, the battle of Chesapeake, 1781, 43 ships with a total of 1 952 guns met in a 2 hour battle, resulting in damage, but no ships captured or sunk. The British scuttled the HMS Terrible 6 days later due to damage, but otherwise both fleet sailed on. It is the same story with the Battle of Hogland, 1788 where Sweden and Russia squared off with a total of 51 ships and 2 416 guns, resulting in a 6 hour battle where both sides captured (not sunk, captured) a Ship of the Line each.

As you can see, while you could kill the crew and damage ships, destroying them completely was hard during this era.

To destroy a ship, the most reliable tactic was to destroy its rigging (by demasting or destroying the rigging), leaving it dead in the water and then turn to rake the ship. This meant that you shot an entire broadside at the front, or even more preferably, the rear of the enemy ship. While the sides of the ship was made of thick oak, the rear were often made up of gilded scuptures and large windows and the timber was far less thick. You could also fire a cannonball through the entire length of the ship, killing, maiming and wounding men, destroying cannon and damaging pumps, rudders, the lower end of the masts etc. When penetrating a side, your cannonball travelled a far shorter distance and could damage far less.

That is why ships of the era sailed in a line (and why ships large enough to be able to do battle under such conditions were called Ship of the Line), to protect each others' front and rear.

This is also why Nelson was such a brilliant commander - instead of sailing in the customary line, he bet that the Franco-Spanish fleet at Trafalgar was not up to notch on its gunnewry due to being in port for a long time and that even if they were, his ships could take the damage - so he sailed in a 90 degree angle towards the Franco-Spanish fleet, letting the first ships (including HMS Victory) in his own column take all the enemy fire and then broke the enemy line and had his ships rake the enemy ships, completely destroying several vessels and shattering the Franco-Spanish fleet's formation, leaving them at the mercy of the Royal Navy.

Adding to the many excellent replies, wooden ships lasted well into the 19th century. One technology that made them dangerously obsolete was the Paixhans gun. It allowed naval guns -- which needed to fire at low angle and high velocity -- to fire explosive shells. The explosive shell was trigged using a delayed fuse, and was embedded in a case.

Prior to the Paixhans gun, explosive shells were fired only from mortars, with high angle and low velocity that made it impractical for naval gunnery.