This is long, but hopefully will be helpful.
Medical licensure is what allows you to practice medicine unrestricted in a given state. It is purely state dependent. Board certification is a national certification designed by national specialty organizations to essentially demonstrate some degree of vetting for your training. Board certification in modern times almost always requires completion of a residency program, whereas licensure simply requires some designated amount of Graduate Medical Education, which again is state dependent. I for example have an unrestricted medical license in the state I'm training in, but I am not eligible for Board Certification, because I have not completed my residency program.
There is technically no such thing as "MD" or "DO" specialty boards, but this is sometimes used colloquially to describe the ABMS specialty boards and the AOA specialty boards. ABMS vs. AOA usually depends on the residency program that you complete, but with the merger, as long as you attend an accredited residency, it will be accredited by the ACGME and so certification will in virtually all cases (save maybe some specific types of fellowships) be through the ABMS. You can opt to do AOA board certification either if you attend a residency with Osteopathic focus or complete Resolution 42 and 56.
Maintaining board certification carries with it certain scholarly requirements that must be completed, including completion of Continuing Medical Education credits, in some cases practice improvement projects, and other Maintenance of Certification tasks. Board certification typically requires passing of specialty board exams. Depending on the specialty, you may need to retake the exam every 7 or 10 years to maintain certification and you will need to complete the MOC requirements at varying times, again depending on your specialty board.
Maintaining licensure does have some requirements, and every state varies with regards to this. Licenses usually last 1-3 years depending on the state, and they renew at a designated time, which again varies depending on the state, but in many cases is related to your year of birth. All states typically require an initial unrestricted license application as well as subsequent renewal applications. Basically all states also require a varying degree of CME as well, but if you are board certified, chances are your specialty board MOC will exceed these. Your license can be revoked or in some way limited by the medical boards if you are investigated for some type of wrongdoing. Also, in many states if you do not explicitly put your license on hold and just allow it to expire, it may have an action against it (most states have a hold/freeze or withdrawal of license option).