The Time Inference with MUSE in Extragalactic Rings (TIMER) project is a survey with the VLT-MUSE integral-field spectrograph of 24 nearby barred galaxies with prominent central structures (e.g. nuclear rings or inner discs). The main goals of the project are: (i) estimating the cosmic epoch when discs of galaxies settle, leading to the formation of bars; (ii) testing the hypothesis whereby discs in more massive galaxies are assembled first; and (iii) characterizing the history of external gas accretion in disc galaxies. We present details on the sample selection, observations, data reduction, and derivation of high-level data products, including stellar kinematics, ages, and metallicities. We also derive star formation histories and physical properties and kinematics of ionized gas. We illustrate how this data set can be used for a plethora of scientific applications, e.g. stellar feedback, outflows, nuclear and primary bars, stellar migration and chemical enrichment, and the gaseous and stellar dynamics of nuclear spiral arms, barlenses, box/peanuts, and bulges. Amongst our first results - based on a few selected galaxies - we show that the dynamics of nuclear rings and inner discs is consistent with the picture in which they are formed by bars, that the central few hundred parsecs in massive disc galaxies tend to show a pronounced peak in stellar metallicity, and that nuclear rings can efficiently prevent star formation in this region. Finally, we present evidence that starbursting nuclear rings can be fed with low-metallicity gas from low-mass companions.