When Time Travel is used to "rewrite" past events, this character will retain their memory of the original timeline—i.e., how everything went the "first time around"—even though that version of events no longer will have happened. Everyone else will only remember the new reality. Examples are Acceptable Breaks from Reality for time travel stories; how can someone Set Right What Once Went Wrong if they don't know that something went wrong in the first place?
This could be justified a number of ways: it doesn't affect time-travelers, the character used Applied Phlebotinum, they are Immune to Fate, they are a Time Master, they have moved to an Alternate Universe, or whatever. The bottom line is that this character has an advantage that no one else does and this is why their memories are unaffected by someone rewriting time. Expect the Timey-Wimey Ball to be thrown around to explain stuff like this. If You Already Changed the Past is involved then the explanation is much simpler; they knew it would happen because they already did it.
The trope becomes trickier when characters who don't have the above justification get Ripple Effect Proof Memory anyway. A Psychic Nosebleed might ensue when someone whose memory isn't completely "proof" gets an "update" on a new lifetime and the mental stress from trying to contain memories from a large number of timelines actually harms the physical body. This might happen even if memories are the only thing that carry over from shift to shift and the time traveler is no longer in his or her original body. The technical term for this is "the time travel clone memory feedback problem." We're working hard to find a cure.
There are two forms of this: in the first the character can remember the previous timeline but has no idea what happened in the altered one until people tell them; while in the second type the character is capable of simultaneously remembering multiple alternate pasts. The latter case can cause Sanity Slippage or outright Go Mad from the Revelation, or at least difficulties in remembering precisely what events happened in which past.
Ripple Effect Proof Memory is inherent in any and all "Groundhog Day" Loop, Mental Time Travel, and It's a Wonderful Plot, and is almost necessary to suffer Time Loop Fatigue. As we already have pages on them, instances of them shouldn't be included here. Individuals with a Ripple Effect Proof Memory may be the only ones who recognize a Ripple Effect Indicator for what it is. When Played for Drama, if this is a rare ability then a character with this may feel isolated and lonely, being the only ones who remember the "original" timeline.
The name refers to the "ripple effect" from the Back to the Future films.
Contrast Caught in the Ripple. Compare Flash Sideways, where characters have glimpses of the Alternate Timeline, but are unable to quite pinpoint what they are, or where they are coming from. If a character only has a vague sense of something having changed rather than intact memories, that's Wistful Amnesia.