According to the PMBOK glossary the budget is the "approved estimate for the project or any work breakdown structure component or any schedule activity." In other words, the budget is an estimate of the costs for the project or a part of it, where the costs have been approved. Actual costs can then be tracked against the budgeted forecast to see that the project stays within or near enough to the estimated and approved cost. Most of the related processes are part of the PMI knowledge area of cost.
Cost is one of the key constraints of a project. Ensuring that a project stays within budget, or that deviations are approved in an appropriate manner, is usually a critical project management activity.