This particular lock is a Sargent and Greenleaf. It's typical of most - the details may vary but the principles are pretty much the same and have been for quite a while.
The tab on the lower right hand corner is for a device called a relock some mechanisms have the tab some don't but any good safe has at least one relock and usually two - one internal to the mechanism and one external. You won't find relocks on cheap safes. One is built into this mechanism like on any good mechanism and I'll point it out later. The relock is a device to prevent the mechanism from opening if force is used to try and open the safe sort of like a mechanical fuse. There are lot's of different designs some use a glass link some a spring some other methods. If the relock trips the only way the safe will be opened is by someone who knows how to bypass it (not going to be cheap) or by cutting the safe open (not going to be easy). Dropping a safe could trip the relock so my advice if moving a safe take the door off - on a good safe they lift off pivots and it makes the safe a lot lighter to move anyway. The brass bolt on the left is the key part. When locked it is extended and when unlocked it retracts allowing another mechanism to move (usually operated by a hand lever).
This shows the side the mechanism, the guard ring and the dial. The door thickness varies on different safes so the shaft on the dial comes long on replacement mechanisms and is cut to length when installing. Mechanisms aren't very expensive all things considered and they do go bad or are replaced for one reason or another.
This is what it is and it's very simple - almost disappointingly so - and hasn't changed much in years. At the top on the left is a bent brass arm - that's the internal relock more on that later. When the dial is turned it turns the brass wheel they are directly connected. The brass wheel turns the top wheel and that turns the middle wheel and that the bottom wheel. They are not directly connected and have some degree of rotation until one wheel picks up the next. This has three wheels so the combination is three numbers there are four wheel models as well.
The wheels have gates and when the gates are lined up the lever shown at the bottom drops in and the bolt can be pulled back. The gate is shown at the top.
Here the gates are lined up - although blocked from view by the lever which has dropped into place and the brass wheel pulled the lever, and bolt, back.
This is the view from the dial side. The gates are lined up and the lever can then drop into place.
Detail of the lever.
Detail of the wheels with the gates lined up.
This has three numbers for the combination and like most mechanisms the combination is dialed in 4 turns to counterclockwise stopping on the first number. Turn clockwise passing the second number twice and stopping on the third. Turn counter clockwise passing the last number once and stopping on the last number the second time. Then turn the dial clockwise about 1/4 turn - it will stop when the bolt is pulled back.
The back of the brass wheel has a tab that engages the top wheel. There is a lot of slop when changing direction almost 360 degrees.
The wheels have a tab on the front and back. This is why there is the pass the desired number sequence it allows each wheel to stop in position and the next wheel to turn with out moving the previous wheel.
When dialing a combination if you pass a number - even by just one you must start all over.
