You need to consult with an immigration attorney. The conditions for naturalization are extensive and does the father support the child financially? He is not eligible from your facts for automatic acquisition of citizenship since he is not in the US and not in the physical custody of the citizen parent. Good luck.
Mr. Lorenzon's answer to your question does not establish an attorney client relationship, but rather is meant to share knowledge with the general public. For specific advise on your case, you need to consult one on one with an immigration attorney. Mr. Lorenzon can be reached at 216 573 7322 or at firstname.lastname@example.org. All initial constulations are free.
The rules for acquiring citizenship through this avenue are different depending upon the year when your son was born and rules of physical presence for the father. You need to hire an experienced immigration attorney to review closely your particular facts.
The short answer is, If your child is a US citizen born abroad, you can get him a certificate of citizenship, which you can in turn use in any situation where you need to prove he is a citizen of the United States.
BUT, IT IS NOT CLEAR FROM THE FACTS PROVIDED HE IS A US CITIZEN.
First of all, are you and his father married. It sounds like you're not, but this is important. Different rules apply to children born out of wedlock if the citizen parent is the father. Which brings me to my next point...
Which parent is the citizen, you or your child's father? This is not clear from the information you provided, but it matters if the child was born out of wedlock and the citizen parent is the father. Then you have to prove a bona fide parent-child relationship (which one of my colleagues was referring to when he asked about financial support),
Also, what is the status of the non-citizen parent? Do they not have status in the United States or are they a lawful permanent resident?
Finally, how much time did the US citizen parent spend in the United States before and after the age of 14?
All of these things are really important to the analysis. Unfortunately, unless the child was born in the United States, the citizenship question is never an easy question to answer, but it IS an important question. If your son is a citizen of the United States he does not need a visa or his "papers fixed" in order to live in work in the United States, he just needs to be able to prove his citizenship.
If he is not a citizen, that opens a whole different can of worms.
So, I have to agree with my colleagues. You need to contact an attorney. Many attorneys will answer questions over the phone for free and just set up a face-to-face appointment if they think you need additional help (which they may or may not charge for).