the other day; a few days ago
Refers to a recent past day, similar to 先日 but slightly more casual and often used in spoken Japanese.
Thank you very much for the other day.
I ran into him at the station a few days ago.
Compound of 先 (saki, 'previous') + の (no, possessive particle) + 日 (hi, 'day'). The exact historical derivation is uncertain, but it follows a common pattern for temporal expressions.