Destroyers and cruisers traditionally had two boats. Both were powered by small diesel engines.
They were the Captain's Gig:
And the 26' Motor Whaleboat (this one was sold as surplus):
Boats were used primarily when the ship was anchored somewhere, though in far rarer occasions, they were used for "blue water" transfers between ships in the open sea. They were hoisted onto the ship and lowered by davits. The davits were double armed affairs that held the boats on cradles when they were not used. When in use, the davit would lift the boat up, tilt down so that the boat was over the water, and then lower the boat on pulleys (called "falls").
When the boats were being raised and lowered, only the bare crew was on the boat and they were required to hold onto the monkey lines (ropes with knotted hand-holds). For one end of the boat could slip off a fall and yes, they occasionally did.
The davits were powered by large electric motors which had limit switches to prevent the motors from breaking things. The motors wound and unwound the wire rope winches on the davits. The limit switches, though, were sitting right out there open to the salt air and they sometimes failed. When they did, the winch motors could wind things a little too tight and bend the living shit out of the davit arms.
During good weather, these boats had a three-man crew: A coxswain, a seaman and an engineman. The seaman and engineman handled lines fore and aft. In bad weather or at night, a boat officer was added, usually the junior-most ensigns in the duty section.
The motor whaleboat was pretty straight forward. The coxswain drove the boat from the steering station next to the motor. The motor had a straight shaft that ran right to the prop.
The gig was, comparatively, a maintenance nightmare. The engine was at the rearmost part of the boat. The driveshaft went forward into a "v-drive", which in turn drove the prop shaft. The housing of the v-drive was made of aluminum. There was no way to get to the bottom of the v-drive other than pulling it out. Aluminum corrodes nicely in seawater, which tends to get into the bilges of boats. So what would happen is that the bottom of the v-drive would swiss-cheese itself from corrosion, the oil would leak out of the v-drive and, if that was not caught, the goddammed thing would seize up.
The gig was the captain's boat and it was at his beck-and-call. A considerate captain would let his gig be used as a liberty launch for at least the chiefs and the officers, if not for the entire crew. It was incumbent upon those who were returning to the ship and who were really drunk to pass up on riding in the gig, as captains took a dim view of squids puking their guts out in the gig's cabin.
When boats were in use, a beach party with a radio was sent ashore for controlling the sailors at the landing point. The petty officer or officer there reported to the OOD. Anyone who was really drunk might have to wait for hours there until they sort of sobered up. The beach party also functioned as a security checkpoint, welcoming and screening visitors to the ship. The boats took their orders from either the beach party or the OOD.
When boats were in use, the senior line officer in the boat was in charge, even if an officer was assigned to the boat and even if that senior line officer was drunk on his ass. More than a few drunk lieutenants got into serious trouble after an incident when someone else on the boat was injured. ("Line officer", in this regard, meant that one was in a warfare specialty eligible for command at sea.)
It was common to hire water taxis when visiting foreign ports.
This served several functions. First off, it freed the ships from having to crew and operate their own boats. Second, it provided some work for the local charter boats, which meant there was some more interest in having naval ships visit. Third, because they were foreign vessels, the ship's officers were not responsible for safety of the water taxi.
Anchoring out was done in ports that either had limited pier/dock/wharf space (or the port wanted to reserve the space for freighters and cruise ships) or were too shallow for the ships to pull in. Most everyone hated anchoring out and using the boats. It was a strain on the duty sections. Boat crews were required to wear the uniform of the day, which meant trashing a set of whites in the summer. Boating operations could be hazardous, especially if the weather was up. Boating might be secured, which meant that you could find yourself stuck on shore for awhile.
A wise sailor on liberty made sure that he or she had enough money set aside to rent a cheap hotel room in case the weather soured. Worse case was when the weather really soured and the ship had to get under way to the relative safety of the open sea. It was not unheard-of for half of the crew to be stranded ashore for a few days, an event that would involve the local consulate/embassy to help out in caring for the strandees.
Carriers and other large ships had personnel boats that were much larger. The admiral commanding a task group would also have his own boat, known as "the admiral's barge."
Working uniform might be authorized if the weather was getting lousy, but that was not to be counted upon.
 "To secure" in the Navy meant "to end an evolution and tidy up." In the Navy, "to secure a building" meant to sweep down the halls, empty the trash, turn out the lights and lock the doors. To the Army, "secure the building" meant to post a guard at the front door. To the Marines, "secure the building" meant to attack the building, blow a hole in the side, go in and kill or capture everyone inside. To the Air Force, "secure a building" meant to negotiate and sign a lease for the building.
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