With the aim of exploring the properties of the class of FR 0 radio galaxies, we selected a sample of 108 compact radio sources, called FR0CAT, by combining observations from the NVSS, FIRST, and SDSS surveys. We included in the catalog sources with redshift ≤0.05, with a radio size ≲5 kpc, and with an optical spectrum characteristic of low-excitation galaxies. Their radio luminosities at 1.4 GHz are in the range 1038 ≲ νL1.4 ≲ 1040 erg s-1. The FR0CAT hosts are mostly (86%) luminous (−21 ≳ Mr ≳ −23) red early-type galaxies with black hole masses 108 ≲ MBH ≲ 109M⊙. These properties are similar to those seen for the hosts of FR I radio galaxies, but they are on average a factor ~1.6 less massive. The number density of FR0CAT sources is ~5 times higher than that of FR Is, and thus they represent the dominant population of radio sources in the local Universe. Different scenarios are considered to account for the smaller sizes and larger abundance of FR 0s with respect to FR Is. An age-size scenario that considers FR 0s as young radio galaxies that will all eventually evolve into extended radio sources cannot be reconciled with the large space density of FR 0s. However, the radio activity recurrence, with the duration of the active phase covering a wide range of values and with short active periods strongly favored with respect to longer ones, might account for their large density number. Alternatively, the jet properties of FR 0s might be intrinsically different from those of the FR Is, the former class having lower bulk Lorentz factors, possibly due to lower black hole spins. Our study indicates that FR 0s and FR I/IIs can be interpreted as two extremes of a continuous population of radio sources that is characterized by a broad distribution of sizes and luminosities of their extended radio emission, but shares a single class of host galaxies.