This is a "lost neutral" and it is very dangerous and should be dealt with swiftly.
Your house has 240V power with "neutral" in the middle. That gives you two sides. The neutral from the transformer forces each side to be 120V. What if it didn't? Then each side would be "whatever" in voltage! (but they would add up to 240V.) That is happening.
The connectors/terminations are far more likely to be the culprit than open runs of wire.
That "whatever" will change with load. Turn on a toaster on one side, its voltage will drop and the other side's goes up. You're bound to over-voltage both sides and blow almost everything in your house... and that can start fires.
What's more, your neutral may not be at a safe (near ground) voltage anymore. Even scarier, some electric ranges and washing machines attach the chassis of the machine to neutral because they are too cheap for proper grounds. With a lost neutral, those chassis are now hazardous. People have died from this. Recently. Google "electrocuted" and that's usually the story. You can't cure this by turning off the breaker; breakers don't switch neutrals.
Source
Here's how a common panel is set up. Notice the two hot buses (copper) - those fingers that stick into the middle feed the breakers on both sides. Notice the trick so each hot bar serves every other row of breakers. Got it?
I would normally say "Turn the main breaker off NOW and don't touch it until it's fixed" but that will result in a fridge full of lost food, and one may be reluctant to do that. One way to temporarily cope with a lost neutral situation is to choose one of the two buses and turn off or pull every breaker that attaches to it. (on most panels: every other row). Now one phase is irrelevant, you have turned your panel into a 120V panel: if neutral goes out, power simply fails, but nothing goes overvoltage. Now instead of being a spooky goblin that fries things randomly, it's a plain failure. Much easier to troubleshoot. However, when power goes out, your neutral wire may be pulled up to 120V... and formerly benign contact with neutral (like those neutral-grounded appliances) is now deadly. Fix this ASAP.