Debt financing refers to capital infusions by entities that do not take any ownership or equity stake in the company or project. Debt financing is like a loan - in fact, bank loans are among the most common forms of debt financing for projects and companies. Most debt is "private," in that it is held in the hands of a single entity (like a bank) or group of entities, and transferring that debt to another party is time-consuming. Just as with stocks, there is "public" debt that is traded openly. Many corporations issue various types of bonds that can be traded no differently than stocks.
The big difference between debt and equity financing has to do with repayment. Equity financing is essentially a loan that is "repaid" through entitlements to a stream of future company or project profits. Debt financing involves various terms of repayment. In many cases, holders of debt have priority on repayment before holders of equity interests in a company.
The cost of debt is determined primarily by how likely or unlikely the lender is to be paid back. If a project goes into bankruptcy, for example, holders of debt may not earn back their entire investments. (One of the advantages of being a lender is that in the case of bankruptcy, lenders often have a higher priority for repayment than equity shareholders.) Rating agencies such as Moody's, S&P, and Fitch use ratings as a general indication of the riskiness of debt.
Wikipedia has nice overview tables and charts of what the various ratings mean. For our purposes, the long-term column is more important than the short-term column. This page shows returns or "yields" on corporate bonds of various grades. Often times, for each step away from the top rating (AAA, for example), investors demand a roughly 0.25% to 0.5% increase in yield in exchange for that increased risk of default, but this is not always the case. (You may see examples when you click on the link for corporate bond return data.)