Satsuki (皐) was the traditional Japanese name for the month of May, and originally was an abbreviation of "Sanae tsuki" (早苗月), meaning "rice seedlings month."
It is also an extremely common female given name, often (but not necessarily) given to baby girls born in the month of May. More rarely, it can be a surname or a male given name.
However, since the original character for "Satsuki" (皐) is not included in the joyo kanji list of approved Chinese characters, most modern-day Satsukis have to use the modern characters for the month of May (五月), which are more often pronounced gogatsu when referring to the month rather than the name. To avoid this, parents have come up with literally hundreds of creative alternative spellings of "Satsuki," although 五月 or just the simple hiragana phonemic spelling (さつき) remain the most common.