So. Kim Davis is a trendy topic. Those probably were not words most of us would think to write a couple of months ago.
Some explanations for people who haven't followed this closely or imagine themselves to agree or disagree with her.
1) County clerks in Kentucky are elected officials. They're still public officials charged with doing public duties, like providing marriage licenses. One result of this elected status though is that they cannot be fired by another public official (a Governor for example). They have to be removed from office via an election or a process of impeachment that is typically complicated and time-consuming.
2) States could decide they don't want to issue marriage licenses, which to me seems like an idea likely to be less popular than is being imagined by some Christians or anarchists and some number of libertarians, but until or unless they do so, some city or county official will be charged with providing these licenses officially. In this case, a county clerk's office was tasked with this duty. There may be other public officials in the county that can perform this legal duty but they have other duties often (judges for example). Since the state gives out marriage licenses on a fairly regular basis, it is not productive to have this task on an ad hoc basis rather than an appointed office handle the regular business of the state.
3) Court rulings based around the 14th amendment have decided, I believe rightly, that these licenses should be issued regardless of gender in the couples involved. This means that any Kentucky law preventing such licenses between same-sex couples from being issued is invalidated and that such laws are to be enforced as amended. So the licenses must be issued and the court's interest is in gaining compliance with its orders that the state officials no longer be violating the Constitution.
The relevant factor here is that it is the state of Kentucky, not Kim Davis, which is seen to be acting when it refuses to issue a marriage license. Kim Davis does not make or decide state laws as a clerk. She carries them out and as such represents the state in that capacity.
4) The judge who issued the order to provide the licenses lawfully a) wrote in the 6th circuit opinion that was overturned that he effectively did not agree with same-sex marriages and b) gave Davis several avenues to lawfully comply.
Most significantly, she was to be allowed to turn over the duties to another clerk in the office rather than issue licenses she disagreed with herself. She refused to allow another clerk in her office to comply whether they wanted to or not or agreed with her beliefs or not. She imposed her beliefs upon others in her employ. This is not a power any elected official possesses, whether they imagine themselves to or not. This is ultimately why she was found in contempt of court and jailed, not for practicing her own religious beliefs but refusing to allow others to live in accordance with their own interpretations of faith and law and imposing upon them a required action that was deemed to be illegal by the federal court system and out of step with directives issued by the sitting governor of the state to comply with those rulings.
One of the apparently forgotten aspects of Jim Crow was that it was a binding legal obligation upon all citizens to segregate, not merely the beliefs of racists but the laws of racists as well were in force and on display. There were ways around this, but it was officially sanctioned by the state to require certain private behaviors, whether people agreed with those laws or not. If people do not agree with allowing same sex couples to marry, they are allowed to disagree with this, to voice opposition, to protest, to attend church services that reinforce these beliefs, and so on. What they are not allowed to do is prevent this from occurring legally and prevent others in their employ from executing the required public duties of issuing a license.
4a) (Update) It appears her dispute is that her name has to be on it as though this is constituting a personal endorsement. It is extremely unlikely that federal courts and judges will decide to amend a state law as to whether her name has to appear on a public form from a county clerk's office as a basis for official documentation being deemed valid. Their concern will be the relevant constitutional status of marriage law enforcement more broadly and not the precise mechanics that the Commonwealth of Kentucky decides to use to provide those marriage licenses, who has to sign where, etc. State laws may provide some protection if she were to sue in state court that some accommodation could be imposed or the state could act to amend the statute to provide such accommodations as allowing a deputy clerks name to be valid, etc, though that's not guaranteed of course. And she hasn't done that anyway. Suggesting her legal arguments are less about seeking accommodation and more about something else, or that her legal team is not very good (also plausible). (update to that part, it looks to me unlikely that the state courts would grant some variety of exemption either as the relevant forms only require the endorsement of the "office of", and not the "person of").
5) The absurdity of claiming that people could drive 30 minutes away for a public service required by state law seems to be a point of order as well. What this suggests in logical terms is that Davis does not agree with same sex marriage, but will abide by such marriages as endorsed legally by others in the same official capacity as herself as long as she has nothing to do with it. Except that she won't do that in her own county.
This is very different from say, abortion providers and distance traveled questions and constraints. The state is not in the business of providing abortions. It does not have to mandate that its officials provide them in a reasonable fashion within a legal jurisdiction of a county or major city. The state is in the business of providing marriage certificates. Since the state is in that business, it is required to do so in a non-discriminatory way (roughly speaking, since the state can prevent you from marrying your cousin or sister or father, or an under age child), and more significantly to this question here, the public and elected officials are tasked to do so as well. If they are unable or unwilling to do so, the effective response is that they should be removed from office in order to gain compliance with the legal environment.
Those public officials are free in their private capacity to complain all they want so long as they comply with the law in a non-discriminatory way or make reasonable accommodations to do so. Not issuing licenses at all in a county, without the approval of other city, county, or state officials, and thus requiring people to go to another county to receive a public service they are entitled to legally is not a reasonable accommodation.
6) The wisdom of jailing her to assure compliance with the law has been frequently called into question. I suspect that the intention of her or her lawyers was for her to go to jail for violating the law and that this is why she refused other methods of compliance. This made it unlikely that other forms of compliance would have worked in the interim to prevent violations of Constitutional law and the state's basic obligations from being ignored in the meantime.
As a form of civil disobedience, this is fine. She was willing to pay the consequences. As a form of protest intent on amending the legal environment, I'm less clear on how this would redound to anyone's benefit if they oppose gay marriage and the change in the legal environment. Some are claiming this makes her a martyr. A martyr to an already activated minority is still a losing opinion on this matter. Her actions would have to in some way change people's minds rather than just fire up people who still agree with her. She is not a church, whose practices were not changed by the legal framework of the 14th amendment requirements to civil institutions, and she was being allowed not to perform the ceremonial tasks personally as a requirement of her official duties. I don't see where that changes people's minds in a meaningful way. If anything, more people being aware that what she was actually doing was refusing to let anyone else operate in an official capacity under her charge should suggest that this is not likely to be a very inspiring case.
7) This question of the intersection of religious liberty with gay marriage (both in the public sphere and in the private business sphere) will come up again with a more presentable case. The Supreme Court refused to hear this one on appeal because it was a terrible case easily dismissed on the merits, most likely being used to fund raise by the legal team rather than attempt to actually reform or repeal the legal environment.