Here is the explanation straight from USCIS:
When you sign the affidavit of support, you accept legal responsibility for financially supporting the sponsored immigrant(s) generally until they become U.S. citizens or can be credited with 40 quarters of work. Your obligation also ends if you or the individual sponsored dies or if the individual sponsored ceases to be a permanent resident and departs the United States.
If the individual you sponsored receives any "means-tested public benefits," you are responsible for repaying the cost of those benefits to the agency that provided them. If you do not repay the debt, the agency can sue you in court to get the money owed. Any joint sponsors or household members whose income is used to meet the minimum income requirements are also legally responsible for financially supporting the sponsored immigrant.
Federal means tested public benefits are the following:
Supplemental Security Income (SSI)
Temporary Assistance for Needy Families (TANF)
State Child Health Insurance Program (CHIP)
You will be responsible to ensure the person does not accept needs based public benefits, permanently, for the rest of their life, or until they can be credited with 40 quarters of working in the US or they gain US citizenship. If they don't work or don't apply for citizenship, you are on the hook and the individual can demand support from you and the government can seek reimbursement from you.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Mr Berman's answer above is correct. However, the likelihood of being ordered to pay monetary damages to the federal government in this instance is extremely remote, and in my 13 years practicing immigration law, I have knowledge of only about two cases where a Court actually ordered a Sponsor to pay monies pursuant to their sponsorship for affidavit of support.