Rosemary is a Mediterranean evergreen shrub, long known as Rosmarinus officinalis and reclassified in 2017 into Salvia as Salvia rosmarinus on genetic evidence, a change many sources have not caught up with. The old name means dew of the sea, from ros and marinus, and refers to its coastal habit; it has nothing to do with the name Mary, though a folk etymology attaching it to the Virgin is old and widespread.
Rosemary for remembrance is the tag everyone knows, from Ophelia in Hamlet, and the funerary use behind it is real: sprigs of rosemary were thrown into graves in England as a token that the dead would not be forgotten, and the herb was carried at both weddings and funerals. The custom is documented well before Shakespeare used it.
Whether rosemary does anything to memory is a more interesting question than it looks. There is a small experimental literature, including work on rosemary aroma and cognitive performance, reporting modest effects on measures of alertness and recall, and 1,8-cineole from the volatile oil is detectable in blood after aroma exposure and has been correlated with performance in those studies. The effect sizes are small, the samples are small, and the field has replication problems. It would be overstating it to say the folklore is vindicated; it would be inaccurate to say it has been refuted. The correct description is an open, low-powered literature with suggestive results.
Carnosic and rosmarinic acids make rosemary extract an effective antioxidant, and it is used industrially as a food preservative.