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A spectroscope is a device to study light's spectrum, a practice termed spectroscopy. In the case of modern advanced instruments, the terms spectroscope, spectrograph, and spectrometer are used interchangeably. Spectroscopes are used as lab instruments and telescope instruments.
The term spectroscope was coined for a device that allows you to look at a spectrum, much like a telescope or microscope that also includes a disperser (prism or grating) and a built-in visible scale to assist you identifying the wavelengths of spectral lines. The term spectrograph was coined for devices much like a spectroscope attached to a camera, that focus the spectrum on photographic film, recording it for later study. The term spectrometer was coined for devices that measure aspects of the spectrum, a general term that pretty much applies to either.
Modern advanced instruments virtually always focus the spectrum on a CCD or similar sensor, allowing the spectral data to be transmitted, recorded and analyzed electronically, and all three terms are commonly used for such instruments. The terms may be used for analyzers of any part of the electromagnetic spectrum, but when used for other than visible light, that is often mentioned, e.g., the phrase "X-ray spectrometer".