Morgan County is in northern Alabama, and it was founded in 1818. If you live near the county seat of Decatur, then getting copies of Morgan County public records will be quite easy. Even if you don't, the mail will work just as well as long as you are not in a hurry.
In Decatur, you need to go to the health department office if you want to pick up your records in person. They are usually issued while you wait, as long as your paperwork is all correct. And for paperwork, you need to have filled out the request form with all the information for the records you want. The form is available at the desk, or you can download it before you go from the Alabama Department of Public Health website. There is one form for all records, so make sure you fill out the correct fields.
With the form, you have a fee to pay (which is currently $15 for most records) and you also have to provide one piece of photo ID for yourself. The fee can be paid in cash at the counter but will have to be either a check or money order if you are sending it through the mail. In that case, make it out to the State Board of Health. The cost is paid for the search, which means you are not refunded your fee even if they are unable to find your records.
Morgan County Birth Records
When getting a copy of your own birth certificate, there is generally no problems and you can also easily access Morgan County birth records for others in your immediate family. Outside of your own family though, there are going to be privacy restrictions as long as the record is more recent than 125 years. Only after that can you make a request for any birth certificate for someone outside of your direct family (such as cousins, aunts or uncles).
Morgan County Death Records
Again, privacy laws keep recent Morgan County death records confidential except for members within the decedent's immediate family. In this case, the time period is only 25 years though. So unlike birth records, there are is a large collection of death records available to the public and they can be further researched through the State Archives if your only need a plain paper copy rather than a certified one. The health department can still issue certified copies either way.
Morgan County Marriage Records
These are going to be open to the public with no restrictions at all. So you just need to identify the record you want by the names of both parties and you can get your copy, no matter who you are. These records were collected by the state starting in 1936, but the local Probate court in Decatur should have older records going back much farther.
Morgan County Criminal Records
Criminal records in any county are never open to the public, so you are only permitted to make a request for your own Morgan County criminal records. The forms will require fingerprints so they are very serious about only releasing these to the proper individual. Copies are issued by the Alabama Criminal Justice Information Center, and their website has forms that you can print out. Then just send the request along with $25 by mail.