You are losing a hot from the pole
In North American {or multiphase European} service, When you lose all your 240V* loads and about half of your 120V** circuits, that means you have lost a leg of power. Here is how most North American panels are laid out, if your lost circuits fit the pattern, that confirms it.
That problem may be around the main breaker in your service panel. It may also be inside your meter pan. However it is most likely out at the electric pole or somewhere in between pole and meter. That is the power company's bailiwick and they fix that for free as part of the service.
What's happening is when the 240V air conditioner cycles on, it is unable to run because one of the poles is disconnected. However, a motor trying to start is quite similar to a dead short, and this has the effect of shorting the two poles of power inside your panel. For instance if pole L1 lost power, all the breakers on broken L1 are now getting L2’s power via the air conditioner. This is no good thing!
You can get this diagnosed for free by calling the power company and reporting an outage. They will also treat it as an urgent matter.
Turn off your 240V* loads, this can damage them.
You don't see any bold headlines on this answer because it isn't another problem: same problem but with the neutral wire broken. In that case it would be a true emergency. Your 240V* loads would work, but your two 120V** loads would have either too-high or too-low voltage (the two voltages adding up to 240V* ***). The too-high voltage can melt appliances and start a fire. The advice would be the same, we'd just be screaming it.
* in Europe and Eurostyle service on 5 continents, this number is 400V. In NYC and parts of Central America this number is 208V.
** in Europe etc. this number is 230V.
*** in Europe or NYC, the voltages won't quite add up to the higher number, because of 3-phase weirdness. There, power is supposed to be a triangle with neutral in the middle. This failure causes it to float around.