Boron chloride exists a monomer while in the same group anhydrous, AlCl₃ exists as a dimer?

Boron trichloride is a planar molecule and three covalent bond results due to sp²- hybridisation. 

BCl₃ does not form a dimer. On the other hand, aluminium trichloride exists in the dimeric state (Al₂Cl₆).

In the dimeric state, each aluminium atom accepts a pair of electrons from the chlorine atom of another aluminium chloride molecule and thereby acquires an octet of electrons. In other words AlCl₃ achieves stability by forming a dimer.