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If you are a U.S. Citizen and your spouse entered the U.S. legally on a visa that allows for Adjustment of Status, then you would file the following forms so that he can stay in the U.S.:
I-130 (Petition for family member), I-485 (application for Lawful Permanent Residency), I-765 (Application for work permit), I-131 (Application for Advance Parole), G-325a (Biographic data - one for each of you), I-693 (Medical exam that a certified doctor must fill out), and I-864 (Affidavit of support). You will need to file each form with supporting evidence and appropriate filing fees. You can find these forms at www.uscis.gov/forms.
The supporting evidence that you would file would be birth certificates, marriage certificate, divorce certificates if either of you have been married previously, proof of your U.S. Citizenship, proof of your spouse's legal entry into the U.S., and financial documents to prove your income over the last year at least.
In about 3 to 5 months after filing, the work permit and Advance Parole (travel permit) should be issued. About 5 to 7 months after filing the interview should be scheduled (though now it is taking longer, so maybe closer to 7 to 9 months). If all goes well, a few weeks later the U.S. Lawful Permanent Residency card (green card) should come in the mail.
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