In plain words
Each line on the bank or card statement should match exactly one transaction in the books. Reconciling finds the ones that do not: a deposit never recorded, a charge entered twice, a check that has not cleared yet.
When it is done, the book balance and the bank balance agree, and every difference has a reason written next to it. It is the first check that the books are true, and it should happen every month for every account.
Example
Three lines from a statement
PG&E, $142.87: bill in the books for $142.87Matched
Cal Telephone, $74.36: bill in the booksMatched
Card charge, $34.00: nothing in the booksFlagged
The flagged line goes to a person, who finds out what it was and records it.
In our model
agents
Agents match the lines. Your person clears what is left and signs off the account.
Agents match the lines. Your person clears what is left and signs off the account.