Lenders generally use two different debt ratios to determine how much you can borrow. The short version is that your monthly housing payment (including taxes and insurance) should be no more than 28% of your pre-tax income, and your total debt (including your mortgage payment) should be no more than 36%. The ratio that produces the lower payment is what the lender will use. Many lenders have more generous qualification ratios, but these are traditionally the most common.
The programs vary in what they can offer. In some cases direct financial assistance may be provided to help you pay your mortgage for a short period of time. Payment plans or reduced monthly payments may be offered. However most of the government programs and non-profit organizations will help facilitate some form of loan modification to qualified homeowners. This will provide families time to get back on track by ideally lowering their monthly payment, reducing interest rates or waiving fees.