Form I-1864a, issued by U.S. Citizenship and Immigration Services (USCIS) is, in fact, a contract between two separate parties: the sponsor and the household member. It is an attachment to the main I-864 Affidavit of Support that nearly all family petitioners of immigrants are required to fill out and sign. The household member makes his or her income available to the sponsor to support the intending immigrant. This helps in a situation where the primary petitioner does not have enough income or assets to sponsor the immigrant alone.
For example, let's say the main petitioner is a 40-year old woman marrying an immigrant. Her 20-year old daughter lives in her house and has a regular job. The daughter could sign an Affidavit of Support to help her mother sponsor the immigrant.
The household member also joins the sponsor in being liable to the U.S. government under the original Affidavit of Support. In literal terms, this means that both the sponsor and the household member agree to reimburse any government agencies who provide the immigrant with need-based financial assistance within a certain number of years. The liability is "joint and several." That's legal language for saying they don't just split their obligations down the middle -- instead, either of them can be held responsible for the whole amount, if need be.
A mere roommate cannot sign an I-864a. This obligation can only be undertaken by an actual relative of the petitioner (spouse, adult child, parent, or sibling), who is living in the same principal residence. Or, even if not living in the same house, a relative or other person whom the sponsor has lawfully claimed as a dependent on his or her most recent federal return can sign Form I-864a.
Not all household members will be willing to sign onto such an obligation, however. But in one important situation, it is in his or her own interest. The intending immigrant, if living in the U.S. already, can, in certain circumstances use his or her own income to sign onto Form I-864a.
The intending immigrant can be a household member if he or she is either married to or living in the same principal residence as the sponsor and can show that his or her income will continue from the same source even after becoming a green card holder. (This requirement can be a problem for immigrants who are currently working without authorization.) The green-card applicant can then sign Form I-864a.
Note also that, in some cases where an immigrating spouse or other relation already lives in the U.S., the petitioner need not fill out Form I-864 in the first place. This is true when the immigrant has already earned (or can be credited with) 40 quarters of Social Security coverage. That usually means working in the U.S. for approximately ten years, having been married to someone who did so, or having been the child under 18 of someone who did so. In such a case, the immigrant would need to fill out Form I-864W to claim the exemption.
For more on the Affidavit of Support requirements, see Financial Support You'll Need to Apply For a Family-Based Green Card.