Context. The expansion of H ii regions can trigger the formation of stars. An overdensity of young stellar objects is observed at the edges of H ii regions but the mechanisms that give rise to this phenomenon are not clearly identified. Moreover, it is difficult to establish a causal link between H ii -region expansion and the star formation observed at the edges of these regions. A clear age gradient observed in the spatial distribution of young sources in the surrounding might be a strong argument in favor of triggering. Aims: We aim to characterize the star formation observed at the edges of H ii regions by studying the properties of young stars that form there. We aim to detect young sources, derive their properties and their evolution stage in order to discuss the possible causal link between the first-generation massive stars that form the H ii region and the young sources observed at their edges. Methods: We have observed the Galactic H ii region RCW 120 with Herschel PACS and SPIRE photometers at 70, 100, 160, 250, 350 and 500 μm. We produced temperature and H2 column density maps and use the getsources algorithm to detect compact sources and measure their fluxes at Herschel wavelengths. We have complemented these fluxes with existing infrared data. Fitting their spectral energy distributions with a modified blackbody model, we derived their envelope dust temperature and envelope mass. We computed their bolometric luminosities and discuss their evolutionary stages. Results: The overall temperatures of the region (without background subtraction) range from 15 K to 24 K. The warmest regions are observed towards the ionized gas. The coldest regions are observed outside the ionized gas and follow the emission of the cold material previously detected at 870 μm and 1.3 mm. The H2 column density map reveals the distribution of the cold medium to be organized in filaments and highly structured. Column densities range from 7 × 1021 cm-2 up to 9 × 1023 cm-2 without background subtraction. The cold regions observed outside the ionized gas are the densest and host star formation when the column density exceeds 2 × 1022 cm-2. The most reliable 35 compact sources are discussed. Using existing CO data and morphological arguments we show that these sources are likely to be associated with the RCW 120 region. These sources' volume densities range from 2 × 105 cm-3 to 108 cm-3. Five sources have envelope masses larger than 50 M⊙ and are all observed in high column density regions (>7 × 1022 cm-2). We find that the evolutionary stage of the sources primarily depends on the density of their hosting condensation and is not correlated with the distance to the ionizing star. Conclusions: The Herschel data, with their unique sampling of the far infrared domain, have allowed us to characterize the properties of compact sources observed towards RCW 120 for the first time. We have also been able to determine the envelope temperature, envelope mass and evolutionary stage of these sources. Using these properties we have shown that the density of the condensations that host star formation is a key parameter of the star-formation history, irrespective of their projected distance to the ionizing stars. Table A.1 is also available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/600/A93Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.