When a company is using double-entry accounting, what elements of a given ledger must be equal?

For every debit, there should be an equal and offsetting credit. It is when the debits and credits are not equal or do not offset that your books don't balance. A key advantage of any automated bookkeeping system is that it will police your debit-and-credit entries as they are made, making it far more difficult not to balance.

In the double entry system, transactions are recorded in terms of debits and credits. Since a debit in one account will be offset by a credit in another account, the sum of all debits must therefore be exactly equal to the sum of all credits. The double-entry system of bookkeeping or accounting makes it easier to accurately prepare financial statements directly from the books of account and detect errors. 

normally, all companies use double entry accounting. single entry use only sole proprieters. 

in ledger, there is nothing should be equal, but in trail balance, balancing figure of debit and credit side should be equal.