Algol (β Persei) is the prototypical semi-detached eclipsing binary and a hierarchical triple system. From 2006 to 2010 we obtained 121 high-resolution and high-S/Néchelle spectra of this object. Spectral disentangling yields the individual spectra of all three stars, and greatly improved elements both the inner and outer orbits. We find masses of M A = 3.39 ± 0.06 M ⊙ , M B = 0.770 ± 0.009 M ⊙ and M C = 1.58 ± 0.09 M ⊙ . The disentangled spectra also give the light ratios between the components in the B and V bands. Atmospheric parameters for the three stars are determined, including detailed elemental abundances for Algol A and Algol C. We find the following effective temperatures: T A = 12 550 ± 120 K, T B = 4900 ± 300 K and T C = 7550 ± 250 K. The projected rotational velocities are v A sin i A = 50.8 ± 0.8 km s −1 , v B sin i B = 62±2 km s −1 and v C sin i C = 12.4±0.6 km s −1 . This is the first measurement of the rotational velocity for Algol B, and confirms that it is synchronous with the orbital motion. The abundance patterns of components A and C are identical to within the measurement errors, and are basically solar. They can be summarised as mean metal abundances: [M/H] A = −0.03 ± 0.08 and [M/H] C = 0.04 ± 0.09. A carbon deficiency is confirmed for Algol A, with tentative indications for a slight overabundance of nitrogen. The ratio of their abundances is (C/N) A = 2.0 ± 0.4, half of the solar value of (C/N) ⊙ = 4.0 ± 0.7. The new results derived in this study, including detailed abundances and metallicities, will enable tight constraints on theoretical evolutionary models for this complex system.