Skepticism refers to a number of different things - it can mean simply a tendency to doubt: if, for instance, someone tells you that they were abducted by aliens or that they had supernatural powers, you would be quite right to be skeptical and ask them for evidence and reasons to believe their claims. If the person were to present evidence of their claims, the skeptic in the everyday sense of that word would be happy to accept it. But a person who subscribes to philosophical skepticism is different: they question or reject the possibility of knowledge more generally, rather than the knowledge of one particular person or claim.
Philosophical skepticism goes back as far as the ancient Greeks, with the argument put forward by Agrippa. The argument is given to us in Outlines of Pyrrhonism by Sextus Empiricus and attributed to Agrippa by Diogenes Laertius. It goes like this: imagine a scenario where a person is putting forth a claim to knowledge of something. That claim must be justified in some way. The justification will usually be another claim. If no justification is given, then the claim is being held dogmatically - "It just is!". If a justification is given, that justification is itself a claim and needs to be justified - this either ends with circularity, where a conclusion of a claim is used as justification for that claim (through more or less layers of indirection), or with an infinite regress, or with a dogmatic basic belief or first principle. All three choices are rather unpalatable: it leaves our beliefs as either unsupported, supported by an infinite regress or supported by circularity. Those who believe that the infinite regress is the least problematic options are called Infinitists, while those who accept circularity are called Coherentists. Foundationalists accept that we get back to one or more basic beliefs. Descartes is an example of a foundationalist: cogito ergo sum (I think, therefore I am) is a mental basic belief upon which, for instance, the general reliability of sense-perception could be based.