We have studied by $^{29}\mathrm{Si}$ NMR the complete condensation kinetics in the conditions of rapid hydrolysis (acidic medium, water in excess) of three silicon alkoxides. The gelation of the tetravalent tetraethoxysilane (TEOS) takes several weeks, whereas the trivalent methyltriethoxysilane (MTEOS) and vinyltriethoxysilane (VTEOS) do not form gels. From a quantitative analysis of the data, we deduce that the first steps of the condensation proceed by progressive assembling of small organized units. This accounts for the very slow kinetics (logarithmic function of time), the occurrence of highly condensed agglomerates, and the absence of gelation in trivalent systems. For the tetravalent TEOS, this is followed by an aggregation phase, which has been studied both by NMR and small-angle x-ray scattering. The fractal dimension D=1.9 and the growing kinetics (cluster size increasing as a linear function of time) are consistent with reaction-limited cluster-cluster aggregation with preferential reactivity at the external cluster sites. Finally, we suggest that the progressive transformation of the sol phase into the gel phase after the gel time can be observed by comparing static and magic-angle-spinning NMR spectra.