Short answers, pulled from the story.
A pseudonym is a part-time fictitious name used only in specific contexts, while a legal name change permanently replaces a person's original name. In many countries, including common law countries, a new name can be ratified by a court and becomes the person's legal identity, whereas a pseudonym coexists alongside the original name.
The Brontë sisters used pen names to conceal their gender in a male-dominated literary profession and to prevent local residents from recognizing their neighbourhood in the stories. Charlotte published as Currer Bell, Emily as Ellis Bell, and Anne as Acton Bell.
A collective pseudonym is a single fictitious name shared by two or more people, typically co-authors. James Madison, Alexander Hamilton, and John Jay collectively used the name Publius for The Federalist Papers. Frederic Dannay and Manfred Lee wrote under Ellery Queen, and the name Carolyn Keene is used by multiple authors of the Nancy Drew series.
Romain Gary published novels as Émile Ajar to test whether his new work would succeed on its own merits, without the benefit of his established reputation. The jury for the Prix Goncourt awarded the prize to Ajar without knowing Ajar and Gary were the same person.
Satoshi Nakamoto is the pseudonym of the still-unknown author or group of authors behind the foundational white paper on bitcoin. Whether the name represents one person or multiple individuals has never been established.
Online reputation systems rely on pseudonymous identities accumulating trust over time through consistent behavior. If new pseudonyms are freely available, bad actors can discard damaged identities and start fresh, a practice called whitewashing or serial pseudonymity. Research by the University of Cambridge found that pseudonymous comments tended to be more substantive and less insulting than those posted under either real names or full anonymity.