Madison was the second county created in the new territory of Alabama, and is was officially founded in 1808 (a few years after Washington County). It was initially Cherokee and Chickasaw land, and it located at the north end of the state.
Alabama maintains some laws regarding the issuance of any vital records (as certified copies) but for the most part, you should have access to many Madison County public records. Recent records may be harder to acquire but historic ones are usually fine.
You can get the application forms from the Alabama Department of Public Health website, and then either mail them to the Vital Statistics office in Montgomery or take them in person to the health department in Huntsville. Mailing will take a few weeks but you typically get your Madison County public records back within a few minutes if you visit the office in person. There is a fee for each record you search though. You will need to provide personal ID as well.
Madison County Birth Records
Madison County birth records are the most limited of the vital records, with a privacy law in place that prohibits most copies of birth certificates for 125 years. This applies to any birth records in Alabama, actually. The only exceptions are that you can order your own birth certificate, or one for an immediate family member. Each copy has a fee of $15, which is non-refundable if they are unable to find your record. Any records that are earlier than 1908 are likely to not have the childs name listed, only the parents and the date of birth. This can make identification tricky.
Madison County Death Records
Also for $15, you can order Madison County death records for immediate family members if the date of death is within 25 years. After that, the documents are considered public domain and then anyone can get copies of any certificates. As public records, they are also available free of charge through the libraries, State Archives and several online genealogy websites that keep U.S state's vital records databases.
Madison County Marriage Records
Alabama started to keep track of county marriage registrations in 1936 though most counties were already doing so through the Probate courts. So you can either get a copy of a Madison County marriage record through the health department or the Probate courthouse (also in Huntsville). These records do not have any privacy limitations so you can go ahead and order any marriage record you wish. The cost for a certified copy will be $15.
Madison County Criminal Records
It will cost you $25 to get a copy of your Madison County criminal records, and you won't be able to order anyone else's. These are strictly confidential and are not released to anyone except the person of record, and some potential employers who have the right to perform background checks. You can make a request for your past records from the Alabama Criminal Justice Information Center, in Montgomery. There is no county office for this. Their website has blank forms and more detailed instructions on making a request.