Our aim is to present an atlas containing 35 low-resolution spectra that correspond to 19 Type II supernovae. These spectra cover different phases of postmaximum supernova evolution, ranging from ~2 weeks to >~1 yr after maximum brightness. The Canarias Database of Nearby Type II Supernovae contains spectra obtained from two different programs: the first, the ``Supernova Monitoring Project,'' was carried out from 1990 April to 1992 July; the second takes advantage of the ORM Service Time facility and is currently active. In this paper we present the first results, mainly compiled from the Supernova Monitoring Project. Based on observations made with the 2.5 m Isaac Newton Telescope and the 4.2 m William Herschel Telescope operated on La Palma by the Isaac Newton Groups of Telescopes at Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.