FeSe is the focus of intense research interest because of its unusual non-magnetic nematic state and because it forms the basis for achieving the highest critical temperatures of any iron-based superconductor. However, its Cooper pairing mechanism has not been determined because an accurate knowledge of the momentum-space structure of superconducting energy gaps $\Delta_i(\vec{k})$ on the different electron-bands $E_i(\vec{k})$ does not exist. Here we use Bogoliubov quasiparticle interference (BQPI) imaging to determine the coherent Fermi surface geometry of the $\alpha$- and $\varepsilon$-bands surrounding the $\Gamma = (0, 0)$ and $X = (\pi / a_{Fe}, 0)$ points of FeSe, and to measure their superconducting energy gaps $\Delta_{\alpha}(\vec{k})$ and $\Delta_{\varepsilon}(\vec{k})$. We show directly that both gaps are extremely anisotropic but nodeless, and are aligned along orthogonal crystal axes. Moreover, by implementing a novel technique we demonstrate the sign change between $\Delta_{\alpha}(\vec{k})$ and $\Delta_{\varepsilon}(\vec{k})$. This complex configuration of $\Delta_{\alpha}(\vec{k})$ and $\Delta_{\varepsilon}(\vec{k})$, which was unanticipated within pairing theories for FeSe, reveals a unique form of superconductivity based on orbital selective Cooper pairing of electrons from the $d_{yz}$ orbitals of iron atoms. This new paradigm of orbital selectivity may be pivotal to understanding the microscopic interplay of quantum paramagnetism, nematicity and high temperature superconductivity.