Step 1: Understand melting point trend.
Melting point depends on the strength of intermolecular or interionic forcesIonic compounds generally have high melting points due to strong electrostatic forces
Step 2: Identify nature of given compounds.
LiBr, NaBr and MgBr$_2$ are predominantly ionic compoundsThus, they have strong lattice forces and high melting points
Step 3: Analyze AlBr$_3$.
AlBr$_3$ is largely covalent due to high polarization of Al$^{3+}$ ionIt exists as a dimer Al$_2$Br$_6$ in solid and liquid states
Step 4: Compare intermolecular forces.
Covalent compounds like AlBr$_3$ have weaker intermolecular forces compared to ionic lattice forces
Step 5: Effect on melting point.
Due to weaker intermolecular interactions, AlBr$_3$ melts at a much lower temperature than ionic bromides
Step 6: Compare with others.
LiBr, NaBr and MgBr$_2$ all have strong ionic bonding and hence higher melting points compared to AlBr$_3$
Step 7: Conclusion.
Thus, the compound with the lowest melting point is
\[
\boxed{\text{AlBr}_3}
\]