Cusp
The boundary where one house or sign ends and the next begins. In a computed chart a planet is always on one side of it: positions are exact, so nothing is truly between signs.
A cusp is a starting line. Sign cusps sit every thirty degrees of the zodiac; house cusps sit wherever your house system draws them. In quadrant systems like Placidus, the cusp is the doorway of the house, and a planet close behind a cusp is often read as leaning into the next room.
The folk phrase born on the cusp usually means a birthday near the Sun's move from one sign to the next. Here is the honest version: the Sun occupies exactly one sign at any instant, and a chart computed for your birth date and time settles which one. What cusp-born people often sense is real but different: Mercury and Venus travel near the Sun and frequently sit in the neighboring sign, mixing the flavors.
House cusps are fuzzier in a productive way, since their placement depends on the system and on birth time accuracy. A planet within a degree or two of a house cusp is a fair candidate for reading in both houses, and a good reason to compare Placidus with whole sign.
See also