Hi there, good question (this was asked a week or so ago but was unfortunately part of the April Fool's day, so many of the answers ... smell ... a little).

In general, regular sailors and officers would use the heads that were near the head (front) of a ship. Boxes (on larger ships) or holes (on smaller or more rudimentary ships) with seats were provided, from which waste fell directly into the sea. You can see an example of boxes here in this picture of the Vasa. Interestingly, "head" is now a generic term for "toilet area" on ships and boats.

The captain had his own toilet in a quarter gallery that was accessible from his quarters, which could also be shared with guests, the admiral, etc. In rough weather or in private cabins, sailors, supernumeraries or passengers could also use chamber pots.

Lots of sources for this one, but N.A.M. Rodger's The Wooden World is a great overview.

edit: removed a couple of words that could have been taken out of context

A good secondary source that can answer this question is Brian Lavery's The Arming and Fitting of English Ships of War, 1600-1815.

1. Lee Chains
   What are lee chains? First, what are chains on a ship? Picture the lines that run from the tall masts down to the sides of ship - those lines where you see sailors running up to adjust sails on the yards. Down where those lines, called shrouds, end is a small kind of platform called channels or chains. What is lee? This is referring to the lee side of the ship, or the side of the ship that doesn't have the wind blowing directly onto it. The wind would be blowing away from the ship on this side. This is important. From the lee chains, men would often take the risk of urinating or sometimes defecating (usually the latter required some kind of sling or other assistance), facing the orifice producing waste away from the ship. With the lee wind, the waste would be carried away from the ship. But, there is the risk of the ship's lee side heeling down and dipping the chains under water. Someone could be easily carried overboard this way. On the biggest men of war with three gun decks, small tiny kinds of round houses became features for men (mostly junior officers) to use on the lee side of the ship.

2. The Head The most well known of the ways to use the toilet, jschooltiger already got that one covered. Though it should be noted, that the ratio of holes to men surprise some, since many ships had only one hole per one hundred men or men. The HMS Victory only had six seats for the 800 men who weren't petty officers and above. It's why other areas had use for waste disposal on the ship in addition to the head.

3. Inboard Seats
   On smaller vessels that didn't have room for a roundhouse near the bow (you need a beakhead bulkhead for that) for officers and the sick, seats had to be built next to the rail of the forecastle that had pipes to take the waste overboard or have seats on the railing itself that would overhang the ship so the waste would fall overboard.

4. The Piss-dale
   Made of lead, and shaped somewhat like the urinals we know today, they were receptacles for urine that were fitted on the inside side of the ships and had pipping to take out to the ship's side like a scupper.

5. Quarter Galleries and Roundhouses
   jschooltiger already mentioned the quarter galleries, but for the officers (both junior and commissioned) and occasional the men from the sick bay, on larger ships (mostly those above the frigates) there were additional roundhouses built at the bow near the beakhead on the beakhead bulkhead. They were similar to quarter galleries, but more basic.

6. Chamber Pots, Commodes, where you could
   Officers, Captains, Admirals, and the sick bay are known for having chamber pots or commodes for privacy or convenience. The head, the quarter galley, the roundhouse, the piss-dale, or the lee chains might seem too far to go (or be occupied), or the weather may be really bad, or some might not want to risk injury (especially at night). Those who had pots or commodes used them as a result. For those that didn't, some just went where they could, trying to find where the ship's bilge went or some place in the hold that the waste would collect out the way. But that was looked down upon since it was unsanitary. But it's another reason why the ship's bilge water often stank horribly, since some of it would be human waste. It's also why there are stories of men being practically poisoned by the air given off from the bilge. For a number of men, laziness won or they couldn't hold it or didn't want to risk going out to do their business.

The only other notes I have on etiquette are these:
Privacy standards are quite different from what we would consider private in the modern western world. The circumstances dictated that you didn't get much privacy going to defecate or urinate. They appeared to have just did their business, and dealt with it. I suspect the mindset on privacy going to the bathroom is more akin to what one would see in the general populations of India and Eastern Asia.
The only other note for etiquette is that dawdling or too frequent use of bathroom facilities could be seen as a man trying to avoid/neglect duty. Though I don't think it would have been used as a dodge as much as it is by stereotypical school children today (honestly, the head is not a place you want to dawdle, its not nice, and, as previously mentioned, not much privacy whatsoever, so it would be obvious if you were there and just trying to waste time).