Alternating breakers have power and no power
It is clear that you have lost a phase or a neutral. The reason you are getting strange readings is when you experience a phase or neutral loss it can turn some circuits into series circuits. This causes all kinds of weird things and readings to happen.

Start by turning all of the branch circuit and feeder breakers off. Then check the voltages just below the main breaker. You should read approximately 120V to the ground bar on both phase A &amp; B, and 240V across A &amp; B. If you are reading anything other than that. You have lost a phase. Then you can do the same thing to the incoming lugs above the main breaker. If you get different results than what you just read below the main, then you have a bad Main breaker. If you get the same result your problem is either the utility power or at the meter. 

If you get the same results above and below the Main, then disconnect the ground wire from the neutral bus and take the same readings both to the neutral and to the ground. If you are getting voltage to the ground and not to the neutral then you have lost a the neutral, and it is usually between the panel and the utility power. Do you have an overhead drop? I have frequently found a break in the drop neutral (the bare wire). 

Use extreme caution when taking these readings because you have no real overcurrent protection if you foul up. If you feel uncomfortable then you should bring in professional help.

The bad news is that if it is a main or the meter installation, you will need a professional to do the repair. I would never recommend that a DIYselfer try to make the repair, since it requires a trained person in safety and procedure to perform such a replacement.

The good news is if it's a break in the utility drop. Then they will make the repair. 

Stay safe and good luck.