Step 1: Understanding the Concept:
In quantum mechanics, all identical particles are classified into two broad categories based on their intrinsic spin: fermions and bosons.
Fermions have half-integer spins (e.g., $1/2, 3/2$). They follow Fermi-Dirac statistics and obey the Pauli Exclusion Principle.
Bosons have integer spins (e.g., $0, 1, 2$). They follow Bose-Einstein statistics and do not obey the Pauli Exclusion Principle (multiple bosons can occupy the same quantum state).
Step 2: Detailed Explanation:
An electron is a fundamental particle with a spin of $1/2$. Therefore, it is classified as a fermion.
A photon is a gauge boson that mediates the electromagnetic force and has a spin of $1$. Therefore, it is classified as a boson.
Thus, the statement "electron is a fermion and photons are bosons" is the physically correct categorization.
Step 3: Final Answer:
Option (b) is correct.