Gravitational-wave memory (GW memory or memory burst) is a lingering effect of some gravitational waves: 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 no-return 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. It is thought that pulsar timing arrays might be able to sense a memory burst from a binary SMBH merger.
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.