During the time period I study, bodies would be buried at sea after a short ceremony, often marked by a prayer service and firing a salute if the deceased were an officer or a prominent member of the nobility.

The general method of burial was to sew the body into a hammock with a couple of round shot at the feet, and then slide it over the side. If the ship were carrying a chaplain, he would be in charge of the prayer service, but if not the ship's captain was able to conduct it, often using a prewritten service or a reading from the Bible or the Book of Common Prayer.

Deaths weren't unusual, especially in the Navy, so the ritual would be familiar to most aboard ship. During a battle, it wasn't uncommon for the mortally wounded or dead to be simply thrown overboard to clear the decks, but anyone who survived and died later (or died of natural causes) would have some kind of ceremony.

There's a pernicious myth that the last stitch would be passed through the deceased's nose, but this seems to be an urban legend, as is the idea that the French buried their dead in the ship's ballast. As a side note, I might mention that the idea that dead bodies spread disease is way overblown -- dead people by their nature don't breathe, move around, vomit, or excrete (though they will leak). Decomposing bodies are certainly not pleasant to be around, which is why we traditionally dispose of them with burial or cremation.

There were rare occasions when a body would be preserved for later burial, most famously in the case of Lord Nelson after the battle of Trafalgar -- his body was placed in a cask of brandy. The legend that sailors tapped into the cask to drink the brandy is likely just a legend, though.