I have lived in this apartment for 2 years now and I keep replacing electronics due to malfunction. I notice that every time My fridge kicks in it dims the lights. Same with the air conditioner in the summer. I have replaced two new computers, a pioneer and denon receiver both brand new as well as 2 powered subwoofers. I always use power bars with the switch on them and turn my equipment off when I'm done. I am now looking at buying a power conditioner. The building I live in was built in the 60's. Any ideas from an expert would be gratly appreciated! Thanks.

Sounds like a Neutral problem, If 2 circuits share a neutral(multi circuit)which a black circuit and a red circuit on opposite phase breakers sharing a neutral and that neutral is loose or open, one circuits voltage will go high and the other will go low due to an imbalance of load(different of resistance) and is seen as you describe(1 high and 1 low).
Could be your Main neutral. A good Tech will likely Take voltage test and visual from the Meter, through disconnect, any splices and Panel connections. Then individual circuits as necessary.
Since it is several, I suspect The Large Main Neutral, between the main and the Panel.
I have even seen a Neutral Barely dangling and connecting, or broke open, follow your ariel lines, can't hurt.

I appologize for the above post, that was for if when the refrigerator kicked it, and it made the lights brighter.
An Electrician needs to check the connections and the Buss Connections where the breakers plug onto, could be burnt/corroded.
Voltage test between the Buss and out of the breaker will tell of any resistance.