Whether or not you can add another breaker is based on two pieces of information: 1) whether or not there is a physical slot to put another breaker and 2) whether or not the load of your home/garage exceeds the 125 amp capacity.

The double 15 amp breaker (with one of those marked as a spare) cannot be used without investigation. In general, a double-pole breaker is either used to provide 240V service, or else for a multiwire circuit. Obviously someone without a lot of knowledge has been at work inside this box.

If space in the box permits, I would install a new 20A breaker (with 12/2 wire) for the compressor. Be sure to install a GFCI (ground fault) receptacle

You CANNOT "pair up another breaker" to get enough amperage.

Given your questions, I would recommend a professional electrician be called to straighten out the issue with half of double-pole breaker being disconnected, and your need for a new circuit.

What breaker is being tripped?

The air compressor?
The drill press?
The panels main?

Induction motors pull many times there running current when starting for a brief period.
If the voltage to the motor falls under the starting load, the starting time is increased.

Regular house type breakers are thermal-magnetic.

The thermal allows for brief overloads for things like motor starting, the equivalent of the old 'slow blow' fuse.

The magnetic reacts for major faults at many time the breaker rating, but trips in a few milliseconds.

If the compressor can start with no other load present, but trips its breaker with another load present the voltage is probably falling and extending the starting surge.

This can cause even the thermal trip in the breaker to operate (the existing load has already supplied some of the heat for the thermal trip).

A larger feed to the panel or a change to 240 V for the compressor might be needed (or both).

I would leave only the lighting on the 15 amp circuit and put in two new 20 amp circuits for everything else. (I hate to have all of the lights go out when my equipment trips a CB.)
I'd put the outlet for the compressor on one new circuit and the drill press on the other. The new circuits should be Ground Fault protected.

The box was wired by a licensed electical company when the house was built, and had to pass an electrical inspection. So I feel it probably wired correctly. Maybe i didn't do a good job of explaining this, so I will try again.
There is only one receptacle in the whole garage. I have a multiplex box plugged into to allow for a few more plugs to be available, (so I don't have to keep plugging and un plugging things). Typically only one thing is used at a time anyway. The multiplex box has a reset button feature on it. It does not actually throw a breaker!
If the compressor is left on and it bleeds down and automatically kicks in, the inital pull will kick the reset on the multiplex if I happen to be using something else like the drill press and don't shut it down immediately. Also if it is real cold in the garage the intial pull will somtimes trip the reset on the multiplex by itslef!
On the double 15 with (spare) on it the one half of the 15 is not wired at all. I have looked at it in the past and one half has now wiring coming to it.
There is one more available space (knock out)in the box for a breaker.
However I am trying to figure out how the add up the used amperage in the box. So that I do not overload the recommended 125 Amps for that box.
Is it possible to post a picture to this site? I think what I am calling a double breaker, may be what they call a slim line breaker, 15 over 15 but all within the same breaker!
They are all like that except for the two that are barred together. The double 30amp barred together is for the dryer and the 25amp barred together is for the water heater.

You don't add up the breakers to determine if it will exceed panel capacity. The calculation to see if your service is big enough is more involved and is not really your problem.

Breakers with the handles tied together are almost certainly 240V brakers. The slim braker indeed give you 2 15A circuits in one space (sometimes 20A).

First thing, I'd figure out what it is that is connected to one side of the "SPARE" 15A.

I would suggest you put your A/C on a 20A braker or figure out how to get it at 240V and on the 15A.

Your options are:

1. Put a new breaker in the empty slot.
2. Use any spare breaker that is indeed not held up.
3. Add more of the skinnies if allowable to make mroe space.
4. Move some circuits to a subpanel and use a larger breaker in the original panel to feed the sub.

Just for terminology, the breakers that take up 2 slots and have the bar going across them are double pole breakers. The ones that take up 1 slot but have 2 switches on them are single pole tandem breakers. Those tandem breakers are designed as space savers and you can run 2 separate circuits off 1 tandem breaker. If you have a tandem breaker and are only using 1 side of it, you can create a new circuit out of the other.

As for overloading the box, in most cases, if you add up all the breakers in a box, you'll find the total exceeds the rating of the box. That is perfectly normal and nothing to be concerned about.

Personally, I would run at least 1 20amp line out there for your power tools - and preferably 2 - and be done with it.

I think I understand the situation.

The problem is that in a residential situation, you can't just add up the labels on the circuit breakers to figure out how much power you need. Since it's highly unlikely that all of the circuits will be drawing their maximum at the same time, the total of the numbers on the circuit breakers will frequently exceed the total power being supplied. To correctly figure out whether you need to increase the supply to your house, you need to perform load calculations, which can be done by professional electricians.

However, in your case, I wouldn't assume that running a compressor from time-to-time is going to put you "over the top", and I would just add a 20A circuit with GFCI as recommended earlier. In the rare case that you do exceed your 125A service, your main breakers are there to protect you.

A double-pole breaker (one with two handles tied together) shouldn't be used as you described, with only half of it being used. Whoever removed the other half should have replaced it with a single pole breaker and an empty slot (with the knock out properly sealed). If you replace the old double-pole breaker with a single pole, then you'll have a perfect slot to put in a new 20A breaker for your compressor.

Ok guys I think I got it now. Alot of bad terminology on my part for one thing!
The one that is labled (spare) is what you called a single pole tandom breaker.
Which means to me that to complete what they needed to wire for the house they only needed one side of the tandom and that is why the other side is labled (spare)! I think they used what they had on hand instead of using a slim line breaker!
So here is what I am going to do I am going to take the other side of that tandom breaker and run a new line 15 amp line for (electical service/receptacles) to the garage.
I am also going to use the knockout left at the bottom of the box and run another new line to the garage. On it I will put in a GFI 20 amp breaker and dedicate it for the air compressor. That should solve my lack of electrical problems!
The one thing I couldn't get past in my head was, no matter how I added up the breakers, it seemed to exceed the 125 amps. Now I understand that would be max if everything was on at the same time!

Except BILL, they are not tandems in this panel. They are as I pointed out, half sized breakers, or what we call skinnies. You get two independent disconnected breakers in one 1" normal mounting position. Yes the both are on the same leg.

Actually they are not skinnies. I went to an electrical supply today and they are tandems. Since they only needed on side of the tandem to complete their origional wiring they left the other side as a (spare).
Quite possibly someone had already punched the knock out and instead of putting two skinnies in the slot they put what they had on hand which was another tandem.
My incorrect terminology helped to confuse the issue. Thanks to everyone who chimed in on this one!!!!!!