We present the first data release of high-resolution (≤0.2 arcsec) 1.5-GHz radio images of 103 nearby galaxies from the Palomar sample, observed with the eMERLIN array, as part of the LeMMINGs survey. This sample includes galaxies which are active (low-ionization nuclear emission-line regions [LINER] and Seyfert) and quiescent (H II galaxies and absorption line galaxies, ALGs), which are reclassified based upon revised emission-line diagrams. We detect radio emission ≳0.2 mJy for 47/103 galaxies (22/34 for LINERS, 4/4 for Seyferts, 16/51 for H II galaxies, and 5/14 for ALGs) with radio sizes typically of ≲100 pc. We identify the radio core position within the radio structures for 41 sources. Half of the sample shows jetted morphologies. The remaining half shows single radio cores or complex morphologies. LINERs show radio structures more core-brightened than Seyferts. Radio luminosities of the sample range from 1032 to 1040 erg s-1: LINERs and H II galaxies show the highest and lowest radio powers, respectively, while ALGs and Seyferts have intermediate luminosities. We find that radio core luminosities correlate with black hole (BH) mass down to ˜107 M⊙, but a break emerges at lower masses. Using [O III] line luminosity as a proxy for the accretion luminosity, active nuclei and jetted H II galaxies follow an optical Fundamental Plane of BH activity, suggesting a common disc-jet relationship. In conclusion, LINER nuclei are the scaled-down version of FR I radio galaxies; Seyferts show less collimated jets; H II galaxies may host weak active BHs and/or nuclear star-forming cores; and recurrent BH activity may account for ALG properties.