B and V time-series photometry of the M31 dwarf spheroidal satellite Andromeda XXI (And XXI) was obtained with the Large Binocular Cameras at the Large Binocular Telescope. We have identified 50 variables in And XXI, of which 41 are RR Lyrae stars (37 fundamental-mode RRab, and 4 first-overtone RRc, pulsators) and 9 are Anomalous Cepheids (ACs). The average period of the RRab stars (<Pab> = 0.64 days) and the period-amplitude diagram place And~XXI in the class of Oosterhoff II - Oosterhoff-Intermediate objects. From the average luminosity of the RR Lyrae stars we derived the galaxy distance modulus of (m-M)$_0$=$24.40pm0.17$ mag, which is smaller than previous literature estimates, although still consistent with them within 1 $sigma$. The galaxy color-magnitude diagram shows evidence for the presence of three different stellar generations in And~XXI: 1) an old ($sim$ 12 Gyr) and metal poor ([Fe/H]=$-$1.7 dex) component traced by the RR Lyrae stars; 2) a slightly younger (10-6 Gyr) and more metal rich ([Fe/H]=$-$1.5 dex) component populating the red horizontal branch, and 3) a young age ($sim$ 1 Gyr) component with same metallicity, that produced the ACs. Finally, we provide hints that And~XXI could be the result of a minor merging event between two dwarf galaxies.
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