My soon to be husband wants to adopt my son. My son's father is not on the birth certificate. What is the process of adoption in this scenario?

Same as in any step parent adoption, though the laws and procedure varies by area.

Generally you have to be married for a period. Once that time requirement is met, your husband petition the local court for the adoption. The bio father will need to be identified and contacted for his approval (doesn't matter that you left him off the birth certificate, that just makes it harder). If you don't know where he is you will have to show the court you made a good faith effort to contact him.

I would not recommend doing this without an attorney. So you can start shopping around for an attorney now.

I might add that his father is only semi present, he will occasionally show up from a drug binge at his parents house, sometimes when my son is visiting... He is not a citizen of our country, and he will be going to jail for a few years?

... The bio father will need to be identified and contacted for his approval ...

The biological father needs to be given notice of the adoption in most states. Frequently, under these circumstances he won't be required to approve the adoption; he merely has to be given notice of the adoption. In other words, an adoption can take place without his approval.

Knowing where the bio father is can work in your favor at least a little because it makes notifying him easier. Adoptions are a very tricky thing that I would never ever recommend doing with out an attorney. Start calling around to find one now that way by the time you have been married for the time that your state calls for everything is in place. Or at least as much as you can have in place is ready to go to court.