Thank God our power came back on line since down Monday, but we only have some circuits in the house with power, I reset all breakers to on but some are still dead. How can there be partial power? How should I begin to troubleshoot?

Sounds like you have only partial power from the utility. The service has two hot legs from utility, you may only have one, as the utility still has problems in your local area.

You may have a problem with your panel, try resetting your Main breaker, and all branch breakers.

To confirm you have partial power , need a voltage tester or meter to test across both hot legs connected to your Main Breaker, should read 240 volts +/-.

Then test from each leg to the neutral bar, (where all white conductors connect), should read 120 volts+/- on each leg.

If you do not have a voltage meter, or if your do not have the knowledge, experience or skills to work on electrical equipment, you should call in an electrician. Working in a live panel requires the proper safety equipment, such as fire rated safety glasses or face shield, voltage rated gloves, etc. A short in a panel can create a fatal arc blast.

You may try calling the utility to see if they can visit to check their equipment .

One thing you can do in this situation is go outside and look at your service drop from the power pole (this is if your power from the utility comes in on an overhead obviously)

This video is about replacing the fused cutout on a power pole, at the very beginning and the end of the video clip they show it installed on the pole next to the transformer.
Look for this outside and see if it is hanging down, that means it is"open" , communicating this to the utility company if you can get through to them on the phone will help them correct the problem sooner.
There are two of these for single phase residential services.
UNDER NO CIRCUMSTANCES SHOULD YOU ATTEMPT TO CORRECT THIS CONDITION YOURSELF, EXTREMELY DANGEROUS

http://www.tdpowerskills.com/node/38