Form I-864A, Contract Between Sponsor and Household Member, is for use in only a limited type of green card (lawful permanent residence) applications: Specifically, it should be used in cases where the U.S. citizen or lawful permanent resident who is petitioning for a family member to immigrate does not have sufficient income to serve as the sole financial sponsor, but needs help from someone living in his or her home to meet the required 125% of Poverty Guidelines level.
You can get a free download of Form I-864A from the website of U.S. Citizenship and Immigration Services (USCIS).
By signing Form I-864A, both the main sponsor and the household member agree that, while the main sponsor will still fill out a Form I-864 and take some financial responsibility for the immigrant(s), the member of the sponsor's household will also provide some financial support to the immigrant(s) if needed. (In fact, under certain circumstances, the household member could become solely responsible for the immigrant's support.)
Both the sponsor and the household member(s) will need to sign the I-864A. The sponsor will attach Form I-864A to the main Form I-864 as part of the immigrant's green card application.
If the household member whose income is being counted is also the immigrant him or herself, who happens to be working legally in the U.S., that person does not need to sign Form I-864A unless agreeing to support immigrating children as well.
Part 1: Self-explanatory; calling for basic name and contact information, to be filled in by the household member. "Relationship to sponsor" means the household member's family connection to the petitioner who is bringing in the immigrant(s).
Part 2: This part is to be filled in and signed by the main sponsor, who will be filling out the Form I-864 Affidavit of Support. Self-explanatory.
Part 3: This is to be filled in and signed by the household member. Self-explanatory.