Gravitational-wave memory (GW memory or memory burst) is the phenomena of a gravitational wave after which things "don't return", e.g., the displacement of test masses by the passing of the wave (the gravitational wave strain) persists. A gravitational wave might have this effect, or might return things somewhat to as before but not completely. Such a memory effect can be linear or nonlinear.
Some detectors are more likely to "see" the non-memory effects of gravitational waves.
Memory burst is sometimes shortened to burst in discussion of GW memory, but the phrase burst gravitational wave is also sometimes used to indicate short, unexplained events of one or a few waves.