First and foremost, almost anyone you run into that happens to call themselves a skeptic will likely be what's known as a empirical (or scientific) skeptic, that is they question the validity of claims that lack empirical evidence. As far as I know, this line of thinking has much of its roots in the writings of David Hume, and is spelled out explicitly in his empiricist tome An Enquiry Concerning Human Understanding. Empirical skeptics accept the epistemological view that contingent claims (claims dealing with the way things are in fact) cannot be verified through deduction alone, but can only be instead verified by experience. For example, an empirical skeptic would say that for us to conclude that lead sinks in water is not a product of logical necessity, but instead it is our experience of that fact that leads us to the conclusion that it in fact sinks.
This, however, illuminates a problem, known as the problem of induction. According to Hume, we develop a sense of how things are in fact through induction (deriving principles from specific instances) as opposed to deduction (deriving specific instances from principles), but induction is inherently imperfect as a means of acquiring knowledge. Where a conclusion necessarily follows from its premises in deductive reasoning, that is not the case with induction. This leaves us with an inherent uncertainty in the validity of all contingent claims. Going back to our example, we must conclude that just because our experience is that lead sinks in water does not necessarily mean that that is always the case. The descriptive principle, "lead sinks in water," is tentative and by no means certainly true. In the empiricist's view, however, the more a principle is confirmed by experience, the more certain we can become of its validity. We can reasonably expect that lead will sink in water because it has always done so in the past.
So, in this context, an "extraordinary claim," as opposed to being a mere claim to the unusual, is the complete opposite of this sort of empirically supported claim. It is a claim that isn't confirmed by experience, and may even be contradicted by it. The claim, "lead floats in water," is an extraordinary claim. That is not a matter of subjective opinion, but a statement that our cumulative experience, mainly our extensive understanding of physics, contradicts this claim. Disregarding trickery or other extraneous variables, lead sinks in water every time anyone tries to test it.
So, when a skeptic says that Jesus rising from dead is an extraordinary claim, we are actually pointing to the body of evidence produced by the study of human physiology, as well as our cumulative experience that people stay dead after being dead for three days.
The next step is to establish what an empirical skeptic means by “extraordinary evidence.” Extraordinary evidence is empirical evidence that meets the rigors of the scientific method. There are a few basic criteria for this type of evidence: it must be testable, falsifiable, and robust. The first two are so intertwined that they are hard to separate from one another in an explanation.
The concept that ideas need to be tested in order to be justified is the foundation of the scientific method. For, if you can not test an idea, you simply cannot determine if it is true or false. To believe in something without putting it to a proper test is to simply take it on the claimant's say-so.
A proper test requires that we are able to distinguish between a world where a claim is true and a world where the claim is false. Thus, the principle of falsifiability comes into play. If a claim is falsifiable, that means that there is a possible test for which, if certain conditions are met, that the claim can be discredited. For example, the claim that "all men are immortal," is easily falsified through the observation of a single dead man. If, however, a claim is unfalsifiable, then it cannot be tested. For, a world in which an unfalsifiable claim is true is completely indistinguishable from a world in which it is false. Thus, such claims are entirely worthless, and warrant extreme skepticism.
For example, in A Demon Haunted World, Carl Sagan proposed to his readers that a dragon lived in his garage. When you went to look in his garage, however, he told you that it was an invisible dragon. When you tried to spread flour on the floor to see its footprints, Mr. Sagan explained that the dragon floated. When you tried to detect the heat of the fire that it spewed from its mouth with infrared sensors, he was so kind to explain to you that the fire it breaths is heatless. And, when you finally decided to spray paint the dragon, he told you that it was an incorporeal dragon. In other words, this dragon is completely beyond our detection. Carl Sagan's claim to its existence is not subject to falsification, and to say that such a dragon exists is rather meaningless, because a universe where such a dragon exists is entirely indistinguishable from one that does not.
The last factor in considering evidence is its robustness. When claim is robust, that means it is supported by an entire body of evidence, and contradicted by little or no evidence. This is important because one specific piece of evidence tends to never point to a single conclusion, but instead is readily open to interpretation and could easily fit with several or even dozens of different hypotheses. It requires not just one single piece of evidence to support a claim, but an entire body, so that all are pointing towards the validity of that claim without mutually supporting any others. An example of a lack of robustness would be alien abduction claims. It's certainly an interesting phenomenon, given the consistency of the experiences between different individuals and the powerful (and quite real) emotional responses to them. However, in them we have a complete lack of robustness, because there are other (more terrestrial) explanations that explain the phenomenon just fine. A combination of sleep paralysis (of which I'm a rather frequent sufferer), hypnagogic hallucinations, aliens in popular culture, and perhaps even effects of magnetic fields on the brain, all explain the phenomenon as well if not better than the idea that aliens are actually visiting Earth and taking human specimens in their sleep. The evidence doesn't only point to the extraordinary claim, and thus less the extraordinary claims are much more likely.
What we’ve learned is that the principle "Extraordinary claims require extraordinary evidence" can be, more formally, written as "Claims that are not confirmed by experience, or contradicted by it, require testable, falsifiable, robust empirical evidence in order to justify them." This is the bread-and-butter of skeptical thought, and once you understand and accept this, you are well on your way to becoming a skeptic.