When our ancestors left the old world to come to the new world they often wanted to forget where they came from. Many of them (men) wanted to become American citizens. To do this they had to file papers with the government of the United States. This was done through the court usually on a town level.
The reason I say just men is that until women received the vote they didn't file naturalization for the most part, they just became whatever nationality their husband was.
Naturalization papers can hold some valuable information but there was not a standard form until the Naturalization Act of 1906 was passed. After 1906 the record includes a photo which may be one no-one has ever seen before, that alone is quite interesting.
In the 1800's Massachusetts had a particularly good naturalization form with great information about the age, birth place and dates that apply to your ancestor and include who sponsored them. Connecticut on the other hand is useless except to tell you when they applied.
You will need to check your state to see how useful they can be.
Now where to find them. Check the National Archives to see if they have the ones you are looking for. Some have been donated to them. Then check the State Archives for your state and finally you may actually have to go out to the courthouse where the event happened.
This seems like a lot of work, it is. But if you get what I got from mine of Cornelius Donahue in Massachusetts, his birth date, place of birth and date he arrived you will realize it is worth the effort. Especially with illusive Irish ancestors this can be the key to locating their place of origin in Ireland.
The person who sponsored them can also be a key to relationships.