The algebra defines the symmetry. The representation tells you what kind of physical thing carries that symmetry.

The Lie algebra $\mathfrak{su}(2)$su(2) is like a universal rotation engine. A representation answers: “Rotation of what?”

• Spin-1/2 rep → rotates electron spin states
• Spin-1 rep → rotates vectors in space
• Higher reps → describe higher spin particles