A spectrometer is any instrument used to probe a property of light as a function of its portion of the electromagnetic spectrum, typically its wavelength, frequency, or energy. The property being measured is usually intensity of light, but other variables like polarization can also be measured. Technically, a spectrometer can function over any range of light, but most operate in a particular region of the electromagnetic spectrum.
A spectroscope is a device that measures the spectrum of light. Early versions had a slit, a prism, and a screen with markings to indicate various wavelengths or frequencies; later versions were calibrated to electronic detectors. Although the apparatus Isaac Newton used in his work on the spectrum of light can be considered a crude spectroscope, it is generally recognized that the spectroscope was invented by Gustav Kirchhoff and Robert Bunsen around 1860.
A spectrograph is an instrument that separates incoming light by its wavelength or frequency and records the resulting spectrum in some kind of multichannel detector, like a photographic plate. Many astronomical observations use telescopes as, essentially, spectrographs.