The Bagua (八卦, "Eight Trigrams") is the eight-sector compass map used in Feng Shui to assign meaning to each direction of a building. Each sector corresponds to one of eight life areas (career, knowledge, family, wealth, fame, relationships, children, helpful people) plus a centre.
Short answer
The Bagua (八卦, "Eight Trigrams") is the eight-sector compass map used in Feng Shui to assign meaning to each direction of a building. Each sector corresponds to one of eight life areas (career, knowledge, family, wealth, fame, relationships, children, helpful people) plus a centre.
In depth
In Classical Feng Shui (Compass School / Li Qi), the Bagua is overlaid on a building using the actual compass facing direction of the front door. This produces the practitioner-grade reading: each sector inherits an elemental quality from its trigram (Qian/Heaven, Kun/Earth, Zhen/Thunder, etc.) and a direction (N, NE, E, SE, S, SW, W, NW).
In the BTB (Black Sect) Western adaptation popular in Western interior-design Feng Shui, the Bagua is overlaid on a floor plan using the front door as the entry, ignoring compass direction. Both schools have followers; classical practitioners typically prefer the compass-based method for accuracy.
The Bagua interacts with Flying Stars (the time-based 9-grid) and your personal Kua number to reveal which sectors of your home are most auspicious for you specifically — and which need remedies.