The following additional Information was provided by 

Mr. Bob Stoner, GMCM (SW), USNR (Ret)

These were later  versions of guns. This Mark and Mod was not placed on the Battleship TEXAS.

20MM and 40MM Guns

20mm Oerlikon gun.

The first 20mm Oerlikons used pedestal that was adjustable (to move the gun cradle up and down).  The early marks through the Mk 4 gun used this arrangement.  However, the mount was very heavy, especially when fitted with the gun shields.  It wasn't long however before it was supplemented and then replaced by the Mk 10.  The Mk 10 was a simple tripod design of a fixed height and was much lighter in weight.  Aboard ship it was accompanied by steel gun shields for the gunner and loader.  Aboard submarines and PT boats, the shields were removed to reduce the size, drag, and lighten the gun even more.  Attached is a photo from the U.S. Navy Manual Gunner's Mate 3 and 2 from the 1950's.  The twin mount Mk 24 was an attempt to double the firepower of the 20mm against the Japanese Kamikaze threat in the 1944 to 1945 period.  The Mk 24 had two guns, with right and left magazines (the LH drum had a white zig-zag mark painted on it for the loader's quick recognition).  The twin gun also used a lead computing sight to help the gunner get more hits on the target aircraft.
[NOTE: The post-Vietnam Mk 16 Mod 4 and Mod 5 guns were NOT Oerlikon-type guns.  These guns were AN/M3 or M24 aircraft guns modified for shipboard use and were belt-fed from either a single 385-round box or two 200-round boxes on the mount.  It was replaced by the Mk 38 25mm chain-gun.]

Mk 10 20mm Oerlikon gun.

Mk 24 twin 20mm Oerlikon guns.

20mm Mk 10 and Mk 24 Oerlikon
The 20mm Oerlikon was an inter-World War Swiss development of the World War 1 German Becker 20mm anti-tank gun.
The Mk 10 (single) and Mk 24 (twin) guns were very simple, mechanical guns.  They had a quick-change barrel that allowed an overheated barrel to be easily replaced with a cool one.  These guns were blowback, advanced primer ignition in operation and required greased ammunition.  With blowback operation, the bolt of the gun is not mechanically locked.  Instead, the breech is kept closed by the force of heavy springs.  The Mk 10 and Mk 24 guns used concentric springs that were fitted around the barrel.  It required about 450 pounds of force to cock a 20mm gun (the twin guns were cocked one gun at a time).  To cock the gun, the loader hooked the cocking bar onto the cocking stud that was attached to the guide bar of the gun.  Once hooked, the gunner hopped up and put the shoulder rests under his armpits and used his weight to cock the gun. 
Ammunition was contained in a 60-round, snail shaped drum that mounted on top of the gun's feed throat.  There was a triangular-shaped stand attached to the top of the 20mm magazine ready service box that held 16 loaded drums.  Empty drums were placed on the loading stand, and individual rounds were inserted into the magazine until it was full.  There was a crank supplied for the drum to tension the spring that fed the ammunition to the gun as it fired.  Normal spring tension was about 9 to 11 clicks.  As the ammunition was loaded, it was greased.  Greased ammunition was required because of the gun's method of operation.
As mentioned, the 20mm Oerlikon was blowback operated, advanced primer ignition.  Because there was no mechanical locking of the breech and the breech had to remain closed until the pressures developed during firing had dropped to a safe level, the gun used advanced primer ignition.  Advanced primer ignition means that the firing pin of bolt fires the cartridge primer as the bolt is moving forward under the force of the driving springs.  The power of the springs combined with the inertia developed by the moving bolt is sufficient to keep the bolt closed until the pressures from firing drop to a safe level. 
Unfortunately, this particular method of operation results in very quick extraction of the fired case.  It is so quick and violent that some means has to be added to keep the recoiling bolt from ripping the back end of the cartridge case off.  [The cartridge case expands to seal the breech on firing and requires a brief amount of time to contract so that it can be extracted from the gun chamber without causing a jam.]  The 20mm Oerlikon does not have any form of primary extraction that could loosen the case before the bolt starts to move to the rear.  The purpose of the grease was to keep the cartridge case from sticking to the chamber walls of the barrel and to assist extraction of the fired case.  If the gun was fired without greasing the rounds, a ruptured case was the inevitable result with the fired case stuck in the barrel.
The only problem that I had with the Oerlikon was the magazine.  The 60-round drum was both heavy and awkward.  It was relatively easy to load if the gun was being used as an anti-aircraft weapon.  However, when used in the surface to surface role, it was difficult for the loader to hold on while the boat pitched and rolled and raise to the magazine high enough to reload the gun.  The Oerlikon was a very reliable and robust gun; it was a good compliment for the lighter .50 caliber machine gun.
20mm Mk 16 Mod 5
The Mk 16 Mod 4 and Mod 5 20mm guns were converted AN/M3 and M24 aircraft guns.  The original gun was developed from the Swiss Hispano-Suiza HS404 cannon developed immediately before World War 2.  The HS404 was used to arm the cannon-equipped RAF Hurricane and Spitfire fighters used during the Battle of Britain in 1940.  Specimens of the HS404 were exported to the United States for manufacture.  These American-built guns became the AN/M1 and AN/M2 20mm aircraft guns.  Although overshadowed by the venerable 50 Browning machine gun, the M1 and M2 guns had some teething problems that were not worked out until the post-war AN/M3 gun.  The AN/M3 gun used lubricated, percussion-primed ammunition.  AN/M3 guns were the primary armament of many post-WW2 fighters and bombers.  A modification of the AN/M3, the M24, was designed by the USAF to fire electric-primed ammunition because the service felt electric ignition was faster and more reliable.
When the Vietnam War broke out, the old 20mm Oerlikons were getting few and spares were becoming harder and harder to find.  The Navy decided to adapt the AN/M3 aircraft guns for use on small boats and other craft.  This gun was the Mk 16.  The Mk 16 was a belt-fed 20mm gun that used a round both larger and more powerful than that of the Oerlikon.  The gun itself was mounted in a cradle that it could recoil and counter-recoil independent of its mounting.  A mechanical feeder/link stripper was attached to the top of the gun, and as the gun moved aft and forward in the cradle, it would work the feeder to bring ammunition to the gun for firing.  During the Vietnam War, these Mk 16 guns were very common on the Riverine Assault Craft that were converted from former landing craft.
After the Navy pulled out from Vietnam in 1972, it decided to replace the remaining Oerlikon guns with the Mk 16 variants.  The difference between the Mk 16 Mod 4 and Mod 5 guns was the method of the sear's release of the bolt.  The Mod 4 used a solenoid and released the sear electrically; the Mod 5 used a mechanical sear released by a cable.  The gun was secured in the Mk 67 Mod 0 or Mk 68 Mod 0 gun mounts.  The Mk 67 had a 385-round ready service box installed beneath the gun.  The Mk 68 had two 200-round ammunition boxes attached to the gun shields.  The Mk 68 gave the gunner the ability to switch to different types of ammunition by simply swapping the ammo feed chutes to the feeder/link stripper.
However, the Mk 16 guns had several problems that were not entirely solved.  The cases had to be lubricated to function reliably (the same as in the Oerlikon).  Instead of grease, the Mk 16 used LSA lubricating oil.  This oil had a Teflon content and there was a neat chamber lubricator added to the gun that operated during recoil/counter-recoil.  The problem was that it only held enough lube for 200-rounds.  A Mk 67 mount had 385 rounds and the Mk 68 had 400 rounds.  This meant that the gunner had to replenish the oil supply in the middle of a firefight.  Instead gunners simply brushed LSA directly onto the linked ammunition and that took care of the problem.
Another problem was not solved.  The Mk 16 was an aircraft gun adapted to surface use.  The result was that the Mk 16 had lots of lockwire to keep it from vibrating itself apart during firing.  There were also special lock tab washers that were used on fasteners where lockwire could not be used.  The result was it took only a few minutes to strip the gun for cleaning, but it took about an hour to lockwire it together.  The other problem was that the special lock tab washers were always in short supply.  Standard procedure was to use them once and replace them; the reality was that this was not always possible.
The last blow was that the Navy/Coast Guard eliminated the only school for Mk 16 gun maintenance in the early 1980's.  The Mk 16 needed operators that understood its operation and maintenance requirements.  When the school closed, this caused lots of problems for ships and craft with the Mk 16.  The result was that the Mk 16 was withdrawn from service and replaced by either the .50 Browning machine gun or the 25mm Mk 38 Mod 0 (aka M242) "Bushmaster" cannon first fielded on the M2 "Bradley" armored fighting vehicle. 
40mm Bofors gun.
The Mk 1 (twin) and Mk 2 (quad) guns were water-cooled guns that were used on ships.  They were heavy and so small craft such as LSTs, submarines, and PT boats needed something lighter.  The solution was to make a Navy version of the Army's towed M1 40mm Bofors air-cooled single barrel gun. 
The Navy version was called the Mk 3 Mod 0 or the Mk 3 Mod 4.  The former gun was all-manual operation; the later incorporated power drives for elevation and traverse.  The Mod 4 also had a more sophisticated lead computing sight than the simple peep and ring type used on the Mod 0. 

A 40mm Mk 3 Mod 0 Bofors gun aboard PTF-17 at Great Lakes, IL in 1974.

A Mk 3 Mod 4 40mm Bofors gun from the Navy manual showing the power drives and lead computing sight (with backup open sights).

40mm Mk 3 Bofors
The Mk 3 Bofors is a pre-WW2 Swedish design that is probably the best light anti-aircraft gun ever designed.  The gun is simple, robust, and totally reliable.  It was manufactured in twin and quad (water-cooled) versions as well as single (air-cooled) versions.  This gun was unique that it was used by Allied and Axis powers during World War 2.  It has been modified many times.  In U.S. Navy service, the Mk 3 gun is to be found as either the Mk 3 Mod 0, Mod 4, or Mod 9.  The Mod 0 gun's operation requires a crew of at least 5 or 6: mount/gun captain [oversees gun operation], 1st loader [loads ammunition into the loader], pointer [elevates and fires the gun], trainer [traverses the gun], and one or two 2nd loaders [to pass ammunition].  Mod 0 operation is strictly manual.
The Mod 4 gun adds power control to the elevation and traverse.  The pointer has a control that allows him to both elevate and traverse the gun under power.  The mount/gun captain is eliminated.  If the power fails, the gun can be operated manually by from the pointer and trainer seats.  The 1st and 2nd loaders are still required.
The Mod 9 gun retains the power control of the Mod 4 gun but can be operated by one man.  The loader is topped by an ammunition drum that contains the ammunition to feed the loader.  The pointer has total control of the elevation and traverse.  In case of power failure, the gun can be operated in manual mode using a 1st and 2nd loader [ammunition drum must be removed] and trainer along with the pointer.  This gun was used on Special Operations craft during the late 1970s through early 1990s.
The Navy has pretty well phased all models of 40mm Bofors out of service at this time.  Yet, as new systems have come about to replace it, none have approached the simplicity and reliability of the Bofors design.  It is still used by many armed forces all around the world.