According to the principles of chemical bonding and molecular structure:

1. Electronic Configuration: Boron ($B$, atomic number 5) has the electronic configuration $1s^2 2s^2 2p^1$ . In its excited state, one electron from the $2s$ orbital is promoted to a $2p$ orbital, providing three unpaired electrons .
2. Hybridisation: These three orbitals (one $s$ and two $p$) intermix to form three equivalent $sp^2$ hybrid orbitals .
3. Geometry: These $sp^2$ hybrid orbitals are oriented in a trigonal planar arrangement with bond angles of $120^\circ$ .
4. Electron Count: Boron forms three sigma bonds with three fluorine atoms. Each bond consists of a shared pair of electrons, resulting in a total of 6 valence electrons around the central boron atom .
5. Electron Deficiency: Since the boron atom has only 6 electrons in its valence shell, it does not complete its octet, which explains why $BF_3$ is an electron-deficient compound and acts as a Lewis acid .

Therefore, the hybridisation is $sp^2$ and the number of electrons around the central atom is 6.