That you are asking questions likely means that you do not know the answers. If you do not know the answers, you likely should review your facts and options with someone who does know the answers, that is, an attorney.
The effect of a DUI on your naturalization application depends on facts that are not in your post. For example, how long ago was the incident? Did anyone get hurt or any property damaged? Are you still under some sort of court supervision?
It may turn out that the DUI will have no lasting impact on your eligibility to apply for naturalization. However, without knowing your facts, no one will be able to provide you with any information specific to your case. This is why you need to review your facts and options with an attorney.
Unfortunately, or perhaps fortunately, different jurisdictions view DUIs differently. As Mr. Nguyen rightly points out, there are a number of issues with regard to an analysis of your case. Consulting with a local, experienced immigration attorney is recommended.
I have found that, in some jurisdictions, the USCIS office will consider a DUI to be proof that one lacks good moral character and would be ineligible to naturalize for five years, thereafter. Others are more lenient and might not consider it a lack of good moral character and analyze the case based upon the totality of the situation.
I think you should post this as an immigration issue.
As has been stated, there are differences between the various Immigration offices.
The conviction alone is not, per se, reason to be denied status or citizenship. However, depending on the individual facts of your situation it could be cause for just such a denial. Consult an immigration attorney in conjunction with a criminal defense attorney.
I agree with Thuong-Tri Nguyen that it will depend on the facts of the case, particularly how long ago the conviction occurred. You may be asking this question because you were arrested for DUI but have not yet resolved the case. Anytime you are charged with a criminal offense, even a misdemeanor such as DUI or domestic violence, a conviction can have a serious impact on your immigration status and application for citizenship and many other immigration issues. Often in these cases it is important to work with both a criminal defense attorney and an immigration attorney.
Obviously, for a non-citizen it is especially important to seek out the services of an experienced DUI attorney as early in the case as possible. As far as timing issues, the immigration attorney may advise you to delay the application for citizenship until after the DUI case is resolved completely (which includes any probation). If you recently entered a plea, it may also be possible to attempt to vacate the plea if you were not advised when you entered the plea that a conviction could affect your immigration status. Such claims must usually be raised within two years of the conviction date.