Prestige is a voluntary reset. You give up your progress, and the game pays you a permanent bonus for it, so the next run reaches the same place faster and then carries on past where you stopped.
The buildings, levels and currency go. The bonus does not, and it is worth more than what you handed over. Every long-running game in the genre is built on that trade.
The arithmetic, which is the whole reason it works
Take a game where your first run takes sixty minutes to reach a trillion coins. You reset, and the game hands you a permanent 2x to all production.
Now walk the second run. Every building costs what it cost before and earns twice as much, so the wait between any two purchases is half what it was, at every step of the ladder. Not roughly half: exactly half, because the price did not move and the income doubled. You reach a trillion coins in thirty minutes.
The thirty minutes you have left over are the point. You keep buying, at double speed, so one hour of the second run takes you as far as two hours of the first would have. And idle economies grow exponentially rather than steadily, so twice as far along the curve is not twice the number. If your coins were doubling every few minutes, that hour does not end on two trillion, it ends nearer a trillion squared. That is a model rather than a measurement, but it is the shape of almost all of them, and it is why the third reset feels absurd next to the second.
A smaller multiplier does the same thing more slowly. At 1.5x the trillion arrives at forty minutes, and that shorter run still ends on a bigger payout than the last one, which buys a bigger multiplier again.
Two caveats. Some games raise costs as your prestige count rises, flattening this on purpose, and some gate progress behind real time rather than currency, where a production multiplier does nothing.
The genre never agreed on a word for it
There is no standard term, which is the most confusing thing about reading about these games. All of the following mean the same mechanic.
Prestige is the generic name, borrowed from Call of Duty 4 in 2007, where hitting the level cap let you reset to level one for a rank badge. Idle games kept the word and attached a real reward to it.
Ascension is Cookie Clicker's word, and Clicker Heroes' word for its first layer. In Cookie Clicker you ascend, convert all-time baked cookies into heavenly chips and prestige levels, and each prestige level is a permanent +1% to production.
Rebirth is Progress Knight's word, and NGU Idle's. Reincarnation is Realm Grinder's second layer, above a smaller one it calls abdication, and transcendence is Clicker Heroes' second layer. Trimps calls its run reset a portal and then, confusingly, uses the word prestige for a small in-run equipment upgrade that is not a reset at all. AdVenture Capitalist names only the currency, and you reset to collect angel investors. Kittens Game just says reset, and pays in karma and paragon.
Crunch, eternity and reality are three separate stacked resets in Antimatter Dimensions. Layer is the community's word for one step in such a stack, popularised by the Prestige Tree family, which are built entirely out of numbered layers.
Stacked layers
The second thing games do with prestige is nest it. A higher layer resets everything under it, the lower layers included, and pays in a currency only it uses.
Antimatter Dimensions is the reference case. Dimension Boosts and Antimatter Galaxies are small resets you perform hundreds of times inside one run. Infinity resets all of that when your antimatter hits 1.80e308, the ceiling of a double-precision float, and pays Infinity Points. Eternity resets your Infinity Points when they hit the same ceiling. Reality resets Eternities and pays in Reality Machines and randomly rolled Glyphs. Five layers, each one turning the layers beneath it into something you clear in seconds.
Progress Knight does the same thing small enough to see clearly. Rebirth wipes your levels and coins but keeps your maximum levels, and grants each job and skill a permanent multiplier of one plus its max level over ten, so a skill you pushed to 300 comes back at 31x. A second layer, "Embrace evil", resets the max levels too and with them every multiplier you built, in exchange for a currency that opens skills nothing else reaches.
When should you reset
Sooner than you want to. The commonest beginner mistake in the genre is hoarding before a first prestige, grinding a stalled run for hours to squeeze out a slightly better payout.
The reason is in the arithmetic above. Prestige payouts usually scale with something like a cube root or a logarithm of your total, so the last hour of a stalled run adds a few percent to the bonus, when resetting would have re-run that entire hour in twenty minutes. Our Trimps and Kittens Game pages give the same advice independently, because both communities learned it the same way.
The rule to use: watch how long one step is taking. When a building, zone or level takes noticeably longer than the one before it and the next few look worse, the curve has flattened and the run has stopped doing useful work. Reset there. Your first run is not meant to be your good run.
Games that have none of this
Prestige is a convention, not a requirement. A Dark Room has no reset layer and no multiplier, and it finishes in two to four hours. Universal Paperclips is the same, a finite game with a real ending. Being completable is exactly what a prestige game trades away in order to keep growing, so when a page tells you a game has no prestige layer, read that as a statement about length and intent rather than a missing feature.