  Euclid will be the first space mission to survey most of the extragalactic
sky in the 0.95-2.02 $\mu$m range, to a 5$\sigma$ point-source median depth of
24.4 AB mag. This unique photometric data set will find wide use beyond
Euclid's core science. In this paper, we present accurate computations of the
Euclid Y_E, J_E and H_E passbands used by the Near-Infrared Spectrometer and
Photometer (NISP), and the associated photometric system. We pay particular
attention to passband variations in the field of view, accounting among others
for spatially variable filter transmission, and variations of the angle of
incidence on the filter substrate using optical ray tracing. The response
curves' cut-on and cut-off wavelengths - and their variation in the field of
view - are determined with 0.8 nm accuracy, essential for the photometric
redshift accuracy required by Euclid. After computing the photometric
zeropoints in the AB mag system, we present linear transformations from and to
common ground-based near-infrared photometric systems, for normal stars, red
and brown dwarfs, and galaxies separately. A Python tool to compute accurate
magnitudes for arbitrary passbands and spectral energy distributions is
provided. We discuss various factors from space weathering to material
outgassing that may slowly alter Euclid's spectral response. At the absolute
flux scale, the Euclid in-flight calibration program connects the NISP
photometric system to Hubble Space Telescope spectrophotometric white dwarf
standards; at the relative flux scale, the chromatic evolution of the response
is tracked at the milli-mag level. In this way, we establish an accurate
photometric system that is fully controlled throughout Euclid's lifetime.
