The FIRBACK (Far Infrared BACKground) survey is one of the deepest imaging surveys carried out at 170 microns with ISOPHOT onboard ISO, and is aimed at the study of the structure of the Cosmic Far Infrared Background. This paper provides the analysis of resolved sources. After a validated process of data reduction and calibration, we perform intensive simulations to optimize the source extraction, measure the confusion noise (sigma_c = 45 mJy), and give the photometric and astrometric accuracies. 196 galaxies with flux S > 3 sigma_c are detected in the area of 3.89 square degrees. Counts of sources with flux S > 4 sigma_c present a steep slope of 3.3 +/- 0.6 on a differential "logN-logS" plot between 180 and 500 mJy. As a consequence, the confusion level is high and will impact dramatically on future IR deep surveys. This strong evolution, compared with a slope of 2.5 from Euclidian geometry, is in line with models implying a strongly evolving Luminous Infrared Galaxy population. The resolved sources account for less than 10% of the Cosmic Infrared Background at 170 microns, which is expected to be resolved into sources in the 1 to 10 mJy range.