This paper presents the first results of the study of high-energy electron and muon neutrino charged-current interactions in the FASER$\nu$ emulsion/tungsten detector of the FASER experiment at the LHC. A subset of the FASER$\nu$ volume, which corresponds to a target mass of 128.6~kg, was exposed to neutrinos from the LHC $pp$ collisions with a centre-of-mass energy of 13.6~TeV and an integrated luminosity of 9.5 fb$^{-1}$. Applying stringent selections requiring electrons with reconstructed energy above 200~GeV, four electron neutrino interaction candidate events are observed with an expected background of $0.025^{+0.015}_{-0.010}$, leading to a statistical significance of 5.2$\sigma$. This is the first direct observation of electron neutrino interactions at a particle collider. Eight muon neutrino interaction candidate events are also detected, with an expected background of $0.22^{+0.09}_{-0.07}$, leading to a statistical significance of 5.7$\sigma$. The signal events include neutrinos with energies in the TeV range, the highest-energy electron and muon neutrinos ever detected from an artificial source. The energy-independent part of the interaction cross section per nucleon is measured over an energy range of 560--1740 GeV (520--1760 GeV) for $\nu_e$ ($\nu_{\mu}$) to be $(1.2_{-0.7}^{+0.8}) \times 10^{-38}~\mathrm{cm}^{2}\,\mathrm{GeV}^{-1}$ ($(0.5\pm0.2) \times 10^{-38}~\mathrm{cm}^{2}\,\mathrm{GeV}^{-1}$), consistent with Standard Model predictions. These are the first measurements of neutrino interaction cross sections in those energy ranges.