On a larger airplane, how do they manage balance?
Before the main cabin door is closed, the ramp personnel will deliver a cargo sheet that lists


number of checked bags
number of gate checked bags
number of heavy checked bags
pounds of commercial cargo
pounds of company cargo


With numbers broken down by cargo bin where accurate.

The flight attendant will deliver a sheet with passenger count in zones.

The FMS or EICAS screens can tell you how much fuel is on board.  

From there, some airline have ACARS programs that you plug those numbers into and it will give you weight information (to turn #bags and #people into pounds). Other have worksheets to calculate the weights.  In the EMB-145 we then had to spin a special wheel with the weight information to get the balance info.  This can vary by operator.

Some operators will have the W&amp;B calculated by dispatch and all of the numbers I mention above are put into the computer system by various people and the dispatcher will deliver the W&amp;B and takeoff reference data to the pilots via ACARS. 

What if there is a problem?

Problems can be overweight overall, overweight in the cabin, overweight in the bag compartments or out of balance.

For overweight, remove people or bags as necessary.  If fuel has not arrived and you can do without some, call dispatch to revise the numbers and load less fuel.

If you are out of balance, you move people or bags around.  This was rarely an issue and if it does happen you just ask a passenger from zone X to sit in zone Y for takeoff.