  It is already accepted that Be stars are surrounded by circumstellar
envelopes, which are mostly compatible with a disc geometry in Keplerian
rotation. We aim to obtain a more complete characterisation of the properties
of the circumstellar environment of Be stars that helps to constrain the
theoretical models of the Be phenomenon. We present near-infrared,
medium-resolution spectra of a sample of galactic Be stars with different
spectral subtypes and luminosity classes. We measure different parameters of
the hydrogen recombination lines from the Paschen, Brackett, Pfund, and
Humphreys series, and use them to diagnose physical conditions in the
circumstellar environment. We analysed the equivalent-width (EW) ratio between
Br$\alpha$ and Br$\gamma$ lines and different diagrams of flux ratios. We also
identify lines from He I, C I, N I, O I, Na I, Mg I, Mg II, Si I, Fe I, and Fe
II. Analysing the EW measurements of particular He I, Mg II, Fe I, Fe II and O
I lines, we find that for some lines they correlate with the spectral type of
the star. Particularly, the emission of the O I~$\lambda\,1.3168\,\mu$m line
decreases towards the later spectral types. We present an atlas of 22 Be stars,
that covers a wide infrared (IR) spectral range with quasi-simultaneous
observations. From a detailed analysis, we define new complementary criteria to
Mennickent's classification of Be stars according to their disc opacity. Some
objects in our sample present compact thick envelopes, while in others the
envelope is extended and optically thin. The correlation between the full
widths at half maximum (FWHM) and the peak separation ($\Delta \mathrm{V}$)
versus $V\,\sin\,i$ for the Br10, Br$\delta$, and Hu14 lines reveals that the
broadening mechanism is rotational. The Ly$\beta$ fluorescence is a key
mechanism to explain the intensity of the emission of Mg II and O I lines.
