Indefinite pronouns like one and somebody: one's, somebody's
The possessive of the pronoun one is spelled one's. There are many types of pronouns. Unfortunately, people explaining the mnemonic for remembering the spelling of its sometimes over-simplify and say something like "it doesn't have an apostrophe because it's a pronoun, like his or her". But actually, as already mentioned, there are many pronouns that have possessive forms ending in -'s. The pronoun it belongs to a particular subset of pronouns that have irregular (or at least irregularly spelled) possessive forms.
RiMMER's answer describes yours, his, hers, ours, theirs, its as "standard" possessive pronouns. For some people, it might help to think of this instead in terms of "definite pronouns" and "indefinite pronouns".
The definite pronouns you, he, she, it, we, they have possessive forms that are spelled without an apostrophe even though they end in "s". (Some definite pronouns have possessive forms that don't even end in "s", such as my/mine, her, our, their.)
But the more numerous indefinite pronouns (one, someone, somebody, nobody, another, etc.) take the usual -'s to mark the possessive.
Not all pronouns belong to either of these two categories. As tenfour mentioned in a comment, there's also the interrogative and relative pronoun who which has the irregularly spelled possessive form whose.
How one and one's is different from other indefinite pronouns
The possessive of one (one's) is formed the same way as the possessive of other indefinite pronouns, such as someone (someone's), but it is used a bit differently. For most people, one is consistently used with the possessive form one's. Other indefinite pronouns can (in fact, must) be referenced in some situations with the possessive form of a third-person definite pronoun like his, her or their. Here's an example of what I mean:
We don't use someone's or no one's in this context (i.e., to refer back to an earlier use of someone or no one in the same sentence). However, many people would say the following sentence:
- One does not like to have one's word doubted.
—A Dictionary of Modern English Usage, by H. W. Fowler (OUP's "Classic" First Edition)
The linked passage in Fowler mentions that "One does not like to have his word doubted" also existed in his time as a competing form. I don't know to what extent modern writers use his (or perhaps their) in sentences like this, but it's definitely less common than one's.
Another way one is unlike the other indefinite pronouns is that it has a special reflexive form, oneself.