I've labeled your photo, which might help you understand what's going on.
The Panel Rating
The panel is rated to support 125 amperes, when connected to a 120/240 volt 3 wire system. This means that 125 amperes can flow through each of the upper bus bars and each main lug, without anything melting or catching fire.
The Upper Section
The top two double pole breakers are before the "main" disconnect, which means they will always have power when the wires feeding the panel have power. I'm guessing there is a disconnect ahead of this panel, maybe at the meter or as a standalone disconnect. Based on their size, I'm guessing one (50A) is for an electric stove, electric heater, or maybe a subpanel. The other (30A) is likely for a dryer, water heater, subpanel, or some other appliance.
The Lower Section
The next double pole breaker (50A) controls the flow of electricity to the lower section of the panel. Unlike the double pole breakers above, this one should not have any terminals where wires can connect. The lower section is rated for a maximum of 100 amperes, so the breaker protecting it must be 100 amp or less (50A in your case).
Branch Circuit Section
The lower section (highlighted in purple) is where the branch circuit breakers connect, and has a maximum breaker size of 70 amperes. This means the largest breaker that can be connected in this section, is a 70A breaker.
Upgrade Required?
Since there is only a single available slot (3), it's not likely a central A/C system could be connected without moving things around (at the very least). Upgrading the panel might be your only option, but it would depend on the existence (or lack thereof) of a sub panel, and the existing service provided to the building (and availability of services in the area).
A subpanel may be an option, however, it would require more information than you have provided here.
Depending on the service provided to the building, upgrading the panel may include an upgrade to the service. Installing a new 125A panel is useless, if you don't also have the service upgraded to support 125 amperes.
To determine if an upgrade is required, contact a local licensed Electrician to do a load calculation on the building.
Breakers Protect Downstream Wiring
Short-circuit and/or overcurrent protection devices (breakers, fuses, etc.), are designed to protect the wiring downstream (after them in the circuit). For example. If your panel did have a main breaker, it would be sized to protect the panel wiring. The breaker would not be sized to protect the wiring feeding the panel or anything before the breaker, only the wiring after the breaker.
Summing Up Breaker Ratings Means Nothing
If you total up the rating of the breakers in the branch circuit sub-section, you'll find that you have 85A on leg A, and 105A on leg B. Which means... Well, absolutely nothing. The only limitations here are that there can only be as many breakers as will physically fit, and no one breaker can be larger than 70A.
If more than 15 amperes flow through the breaker in slot 4 , that breaker will trip (open). This protects the wiring connected to that breaker. If more than 50 amperes flow through any combination of breakers on either leg A or B in the branch circuit sub-section, the sub- section breaker will trip (open). This protects the wiring between the sub-section breaker, and the branch circuit breakers.
You could theoretically. have breakers totaling 1,000,000A. It still wouldn't matter, as long as you have proper overcurrent protection.
If I've missed anything, or haven't explained something properly. Feel free to ask additional questions, or point out mistakes in the comments below.
Best Answer
Ok first we don’t add values up like that but a cheater, tandem or double stuff is rated for the amperage on the handle so yes that cheater can draw up to 40 amps without tripping on that one leg 20 at each breaker. What if you put a 60 amp breaker in that panel that and a 40 and 4 20’s , it is possible it would be just fine we add up the loads on the circuits and in some cases may not count one, for example the 60 for an electric heater and a 40 for a AC unit you would only count the 60 the 40 not running the same time as the 60, this is all part of load calculations your 15 amp breaker will rarely have 12 amps on them when they do have that or 15 it is usually for short periods of time. These are additional safety’s built into code and the diversity of the loads makes a difference SO is your box overloaded? Not even close you really could have much larger breakers as my examples tried to show you are probably drawing just about 1/2 the rated current of this sub and your limiting factor is the breaker feeding it if you have a 125 amp breaker feeding this sub it will protect the panel.
You have a circuit that is all 12 gauge wiring and would like to bump the breaker from a 15 to a 20 go ahead , at some point if you start tripping your main you have gone two far but I rarely see that with all small breakers and a few tandems . As long as your panel is rated for tandems it is fine the buss can handle the load. If they are true cheaters (non current limiting) devices you may overload that buss location but I doubt it with a 15 full size on the other side.