A systematic study is presented on the superconductivity (SC) parameters of the ultrapure niobium used for the fabrication of the nine-cell 1.3 GHz cavities for the linear collider project TESLA. Cylindrical Nb samples have been subjected to the same surface treatments that are applied to the TESLA cavities: buffered chemical polishing (BCP), electrolytic polishing (EP), low-temperature bakeout (LTB). The magnetization curves and the complex magnetic susceptibility have been measured over a wide range of temperatures and DC magnetic fields, and also for different frequencies of the applied AC magnetic field. The bulk superconductivity parameters such as the critical temperature T c ¼ 9:26 K and the upper critical field B c2 ð0Þ ¼ 410 mT are found to be in good agreement with previous data. Evidence for surface superconductivity at fields above B c2 is found in all samples. The critical surface field exceeds the Ginzburg-Landau field B c3 ¼ 1:695B c2 by about 10% in BCP-treated samples and increases even further if EP or LTB are applied. From the field dependence of the susceptibility and a power-law analysis of the complex AC conductivity and resistivity the existence of two different phases of surface superconductivity can be established which resemble the Meissner and Abrikosov phases in the bulk: (1) ''coherent surface superconductivity'', allowing SC shielding currents flowing around the entire cylindrical sample, for external fields B in the range B c2 oBoB coh c3 ; and (2) ''incoherent surface superconductivity'' with disconnected SC domains for B coh c3 oBoB c3 : The ''coherent'' critical surface field separating the two phases is found to be B coh c3 ¼ 0:81 B c3 for all samples. The exponents in the power law analysis are different for BCP and EP samples, pointing to different surface topologies.