Coffee County is in southern Alabama and was founded as a county in 1841. Many vital records will go back at least that far but they can be sporadic and incomplete until 1908. That's when Alabama made it mandatory to register births and deaths across the state.
The most convenient place for you to get a copy of any Coffee County public record is the local health department office in Enterprise. All you need to do is go to the office, and they have while-you-wait counter service during standard business hours. You can pick up the form when you get there, or download a blank one from the Alabama Department of Health website. In total, you will need a completed form, some ID of your own (government-issued with a photo) and the appropriate fee for the particular Coffee County public records you are looking for.
If you are paying by check or money order, make it out to the State Board of Health. These fees are not returned if your records are not located, so make sure you are providing as much accurate information as possible.
You can also mail in your application materials to the state Vital Records department in Montgomery if that suits you better, but expect a 1 to 2 week wait.
Coffee County Birth Records
There is little difficulty in requesting a copy of your own Coffee County birth record, or the records for immediate family members. But other than that, you will only be able to make a request if it's been 125 years since the date of birth. All birth records in Alabama are considered confidential until that much time has passed. If you are looking for such older records, you should contact the State Archives. To get a certified copy from the health department will cost $15 USD.
Coffee County Death Records
Coffee County death records are also subject to a confidentiality period but it is only 25 years long, so it will be easier to retrieve records outside of your immediate family after that time. Until then, the restriction remains to children, parents, siblings and spouses. Getting a certified copy is $15 USD but many older (over 25 years) records can be acquired for free through local libraries, archives and even online genealogy sites because these records are public domain.
Coffee County Marriage Records
The state started to collect Coffee County marriage records in 1936 but you can get records much older if you inquire at the Probate court in Elba. They may (or may not) have additional material holdings earlier than 1936. Otherwise, you can place your order with the health department as mentioned above, and get a certified copy of anyone's marriage record for $15 (it's only $3 if you go the the Probate office).
Coffee County Criminal Records
Criminal records are not accessible like other vital records are, regardless of time period or family relation. Only law enforcement officials and employers may access Coffee County criminal records unless you are requesting your own records. You will have to put in your application with the Alabama Criminal Justice Information Center in Montgomery, and the fee for these records is $25 for each copy.