Budgets are a feature of the new Azure cost management tool, which is primarily the integration of Cloudyn into the Azure portal. Budgets allow you to set a financial boundary for a subscription or a resource group, which you can monitor through cost management and trigger alerts when you are close to the cost set in your budget.
One thing you cannot do with budgets, and something that many people feel is an obvious requirement, is to stop people spending any more money when they hit the budget. If a subscription or resource group breaches its budget, then it will trigger an alert. However, it does not turn off any resources, and it does not stop further spending.