A re-entry is what it sounds like: a player withdraws from a tournament and is allowed to enter it again for a second time. FIDE refers to this as the restart option.
A player who re-enters a tournament has to pay another entry fee; some tournaments offer re-entered players a discount for their return.
(See the glossary for definitions of more chess terms.)
A re-entered player cannot play someone they faced “in their first life, ” unless that opponent has also re-entered. Then, the “re-incarnated” entities can play!
Most tournaments do not allow re-entries, and scholastic tournaments almost never do, but it is something to be aware of.
Are Re-Entries Fair?
I think just about any tournament policy is fair if it is announced in advance in all publicity. It is the responsibility of the player to understand the rules of a competition, and to ask questions of the Organizer or Tournament Director if they are unsure about something.
The Organizer is responsible for ensuring good playing conditions; the Tournament Director is responsible for applying the regulations of the competition correctly and fairly.
Sometimes a re-entered player will win a prize, and this can upset some players. Anecdotally, the re-entry doesn’t change the player’s fortunes and they just increase the prize fund for players in good form.