I have a home that was built in 1957. We turned the central air on recently and now when anything else is running, such as the dryer or the washing machine, the lights dim and flicker and the backup power supply keeps resetting. I do know that the wiring is only 2 wire with no grounding. What is going on? Also power distribution panel has been upgraded to breakers.

You could have some big, expensive problems. Replacing the power distribution panel without upgrading the grounds could have violated code. As long as you leave it alone, anything that meet code when installed is grandfathered in. It doesn't take much change to require bringing everything up to current codes. Was it done since the last time the central air worked without problems? If so, likely something isn't quite right. Call back whoever did the work. It could be a simple, forgivable error if they are willing to correct it at their expense. The problem could also be the incoming power. The electric company may be falling behind as demand in your neighborhood grows. They may be no more quick to admit they have a problem and fix it, than a shady contractor. You may also need to talk to the building inspector for your area.

Your lack of grounds may not be as big of a problem as you think. I think it has always been required that all the outlet and switch boxes are required to be grounded. Today that is usually met by the third, uninsulated wire in the usual NM 12-2 with ground. In the 50's, it may have been met with metal wire housings, either the pipe like steel conduit, or the metal flex commonly know as BX. This is simple to check. You may be able to see the wiring as it comes out of the box, or in unfinished parts of the house. Better yet is to measure between the contacts in an outlet and the screw holding the cover plate on. One contact should show 0 voltage and resistance between it and the screw or outlet frame. The other should show 115 or so volts, or light a test light. If so, likely your light fixtures are already properly grounded. You can upgrade to 3 prong outlets simply by shutting off power to the circuit and removing the 2 prong outlets and transferring the 2 wires to new 3 prong ones. If they let 2 conductor non metallic slip by when your house was built, it could be expensive to fix. It could also be dangerous not to fix it. If one of your appliances with a metal housing shorted out, somebody could get killed. Some electronic equipment works better with a real ground.

This is a problem you need to get fixed. If you have low voltage either because of problems with your wiring, or the incoming power, everything in your house will be running on low voltage. This makes it draw more power, overheating both it and the wiring feeding it. Low voltage leads to premature failure of motors including air conditioners. In some places the regulators and environmentalists do not let the power company expand as they need to. If the problem is low incoming voltage, and the power company will not, or can not fix it, you may want to look at replacing some of your electrical appliances with natural gas or propane starting with the older, and higher capacity ones. Maybe even look at some of the alternative electrical supplies including solar.