Algol (β Persei) is the prototypical semidetached eclipsing binary and a hierarchical triple system. From 2006 to 2010 we obtained 121 high-resolution and high signal-to-noise ratio échelle 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, v_B sin i_B = 62 ± 2 km/s and v_C sin i_C = 12.4 ± 0.6 km/s. 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 summarized 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.