What forms do I file for my non-citizen husband?

I-130, I-485, G-325A for each of you, I-864, perhaps I-864 for co-sponsor, perhaps I-864A for household member of co-sponsor, I-765, I-131 (only if eligible). May be other forms if any issues in inadmissibility. I recommend you use an immigration attorney to handle the process as there is lot more involved than just completing some immigration forms.

You can file for his Adjustment of Status (I-485). There are many other applications that go with an Adjustment and you should consult an attorney to confirm his qualifications.

Generally, a U.S. Citizen may petition for a spouse to adjust status to become a Lawful Permanent Resident (to get a "Green Card") if the spouse entered the U.S. lawfully and with inspection, even if he may have overstayed his visa or worked without authorization. The immigration forms that must be filed include a Petition for Relative Alien, an Application for Adjustment of Status, biographical information forms, the petitioner's Affidavit of Support (if the petitioner's income is insufficient, then an Affidavit of Support from a joint sponsor will be needed too), medical examination results, birth certificates, marriage certificate and evidence of the bona fide nature of the marriage. In some instances, other documents will be needed too (such as an applicant's naturalization certificate, if applicable; divorce decrees for either or both parties, if applicable; certified arrest disposition documents, if applicable; tax returns and sometimes other evidence relating to the petitioner's income; etc.). Proper completion of a marriage-based immigration application can be more complex than meets the eye. Errors or omissions, including failure to properly supply all applicable documents, can lead to very significant delays or even denials (and where the foreign national already is out of status, that presents additional risks). It would be wise to work with an immigration attorney who can review all circumstances relating to you and your husband in detail, and help prepare the most persuasive application package. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.