Why levels of radio contours maps are given in mJy/beam and what does it mean? I know what a jansky is, but I don't understand what beam (in $\frac{1 Jy}{beam}$) is.
If jansky is a unit of flux density, what kind of unit is $\frac{1 Jy}{beam}$?
Jansky is defined as  $1 Jy= 10^{-23}erg/s/cm^2/Hz$. So all the energy coming from a given solid angle per time, per frequency bin and per detector area.

This is convenient for sources of small angular extent (smaller than observing beam), e.g. for point sources, because the flux remains constant for varying beam sizes. For extended sources, the surface brightness is often described with units of Jy per solid angle. Radio contour maps, e.g. of interstellar clouds, usually show objects that are larger than the beam, so that the observed fluxes are a function of the beam size. To clean the beam influence from the date Jy/beam (beam area in steradian) is used. 

So it is a surface brightness. For a flux of 1 Jy measured with a beam of 1 sr solid angle we find:   $1 Jy/beam= 10^{-23} erg/s/cm^2/Hz/sr$. 

Note, that 1 sr is already a huge portion of the sky. Beams are much smaller, on the order of arcsec to arcmin diameters.