Alabama's Jackson County was named after early U.S president Andrew Jackson, and it was formed in 1819.
The county seat is in Scottsboro, which is where you can go to get copies of most Jackson County public records. Specifically, you need to visit the health department office. They issue certified copies of birth, death and marriage records to anyone who makes a request provided you are eligible to have a copy and have the paperwork in order.
A proper request will include a complete form, the search fee and your own photo ID. You just have to show the ID to the clerk, not actually submit it with the forms. The fee can be paid in cash, or by check or money order. Have it made out to the State Board of Health. If they can't find the record you ask for, you don't get a refund.
You can visit the Scottsboro office for while-you-wait Jackson County public records but if that is not an option for you, they accept any mailed in requests as well. Just add a photocopy of your ID to the package.
Jackson County Birth Records
Birth records in Jackson County (and the rest of Alabama, actually) are protected for 125 years, so can't be accessed by just anyone for that length of time. For that time period, only the person on the certificate can request a copy, and so can any members of his or her immediate family. After the privacy restriction is past, you are free to get copies of any Jackson County birth record you wish and you can research them at the State Archives at no cost. Certified copies through the health department are $15 USD each.
Jackson County Death Records
You won't have quite so long to wait for available death records. The privacy limitations only last for 25 years, during which time only immediate family members can get copies of a Jackson County death record. When getting a certified copy, the fee will be $15 but all records after 25 become public domain and can be searched for free if you know where to look. The State Archives has copies, and you can usually use the interlibrary loan service to access them through your own local public library.
Jackson County Marriage Records
Jackson County marriage records are available to the public though you will still have to pay the $15 fee if you want a certified copy through the health department office. The state began requiring counties to collect marriage registrations in 1936 but most were already doing so at the local Probate court. You can usually make requests for marriage records through them if it is more convenient.
Jackson County Criminal Records
At no time are any Jackson County criminal records made available to the public so you are going to be limited in only being able to request your own background files. Employers also have some access in order to perform legal background checks on prospective employees. Your document request needs to go to the Alabama Criminal Justice Information Center in Montgomery, and each request has a fee of $25. Their website has the forms you need.