落款 is the colophon block on a Chinese painting or piece of calligraphy: it usually includes the date, the artist's name, sometimes the recipient (
上款 'top inscription'), and a red seal (
印章 yìnzhāng).
落 here means 'to drop / set down', not 'to fall' — you 'set down' the inscription in the corner. Verifying the
落款 is the first thing experts do when authenticating Chinese art.
