Adults make a plan and follow it, children do what feels good. - Dave Ramsey
Budgeting is the act of making a plan for where money should go before you spend it. It is preemptive accounting, accounting being the keeping of records on where money went after it was spent. The budget should, at minimum, put a target limit on all spending categories for a fixed period of time but it does not need to predict any yellow arrow transactions and it typically does not include green arrow transactions. Budgets are a tool for controlling spending so they need to address spending categories.
A useful budget must be designed by the following criteria:
Budgets are a tool for financial beginners struggling with self-control. Those who cannot seem to help but run their financial lives off a cliff without thinking ahead need to budget. If you are saving more than 25% of your income without a budget (as recorded year by year with real records and not a silly guess), you can live just fine without a budget. Everyone else should have a budget that they stick to religiously.
By far my favorite anecdote about budgeting and accounting illustrates the point that a person who is not coming to appropriate conclusions is a person who does not have a complete set of facts. Dave Ramsey described the largest fight of his marriage over their first budget as a disagreement about groceries, him thinking $500/mo was ridiculously high, and Sharon his wife saying it was conservative and he did not know what anything cost. He went to the grocery store to prove her wrong but lo and behold, she was right and once confronted with solving the problem in reality rather than in theory (otherwise known as "in Dave's imagination"), Dave came around to the same conclusion his wife had.