Number Counts of galaxies in two GALEX bands (FUV: 1530A and NUV: 2310A, both in AB magnitudes) are reported. They provide for the first time in the literature homogeneously calibrated number counts of UV galaxies covering continuously a very wide range of UV magnitude (14 -- 23.8). Both the FUV and NUV counts are inconsistent with a non-evolution model, while they are in good agreement with evolution models (essentially luminosity evolution) derived from the high-z UV luminosity functions of Arnouts et al. (2004). It is found that the contribution from galaxies detected by GALEX to the UV background is 0.68+-0.10 nW m-2 sr-1 at 1530A and 0.99+-0.15 nW m-2 sr-1 at 2310A. These are 66+-9% and 44+-6% of the total contributions of galaxies to the the UV background at 1530A, respectively, as estimated using the evolution models. ...