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 petitioning for a family member 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.
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. 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; filled in by the household member.
Part 2: This part is filled in and signed by the sponsor. Self-explanatory.
Part 3: This is filled in and signed by the household member. Self-explanatory.