Each spectrograph splits electromagnetic radiation (within the wavelenght workspace and the available resolution) in quasi-monochromatic radiation. We work in the visible range, ie at wavelengths between roughly 4000 and 7000 Angstroms = 400 to 700 nm (0.0000004 to 0.0000007 meters). The "white" light consists of a stream of photons (= smallest units of electromagnetic radiation = small localized wave packets, each representing a given energy unit). The photonic energy determines the color = wavelength of the photon.

The most important part of a spectrograph is the dispersing element. It splits the photons of different color. For visible light spectroscopy prisms and optical gratings are commonly used.

A prism is exploiting the difference in refractive power of glass for photons of different energy. For details, see http://en.wikipedia.org/wiki/Dispersive_prism.

Optical gratings work by light diffraction and interference caused of regularly repeating patterns (think of shimmering color CD's). For details, see http://en.wikipedia.org/wiki/Diffraction_grating.

For various types of spectrographs see here.

Some optical basics.

The simplest spectrograph consists of a dispersive element (ie, prism or grating) and a camera (= imaging optics) and a detector. The detector can be your eye, then this is called a spectroscope. For faint objects you will use more sensitive detectors, especially before the photographic plate, now prefer electronic detectors such as CCD or digital cameras.

For amateur astronomers here is the so called Staranalyser (graph left side). This is simply a transmission grid, which is attached to the lens apparatus of a normal photo or a CCD camera. This allows to get low-resolution spectra of all stars in the field of view produced by the camera.

See also http://www.astrosurf.com/~buil/staranalyser/obs.htm
or
http://www.patonhawksley.co.uk/staranalyser.html.

With a Staranalyser can be obtained in the simplest way first experiences with stellar spectra, in particular, will be familiar with the spectral types of stars. The Staranalyser disperses the parallel light beams of the stars incidenting from the sky.
This is generally valid: all dispersive elements such as prisms and gratings need a parallel incident light beam.
Behind the grating the original parallel starlight is deflected, the different angles depends of the color = wavelength. After imaging by the camera the originally starpoint is deformed to a colored stripe.

In order to be able to Faint Object Spectroscopy use a light-harvesting machine. This is the telescope. It concentrates the light falling through its aperture in a small image of the star in the focus. Subsequently, the light of the small (usually 15 to 60 um) star disk in the focus is again converted into parallel light by the collimator. However, this parallel beam is much narrower as the opening of the telescope and therefore more concentrated = brighter (see graph).
Collimator and the telescope should have the same focal ratio. Then the entire cone of light from the telescope, which cuts in the shared focus of the telescope and collimator (top of the cone) and re-opens in the direction of collimator, also fully covered by the collimator without geometric loss of light and transmitted as a parallel beam to the grating or prism is (see Figure, size only to demonstrate the equality of the focal ratios (lens diameter / focal length) of telescope aperture and collimator)
Typical f-numbers lies between 1: 4 to 1:15.

Thus, while the Staranalyser directly works with the parallel incoming starlight (works without a "concentration") the combination of telescope and collimator serves only the light collection and concentration. Strictly speaking, telescope and collimator are one instrument (although the collimator is placed in the spectrograph). In front parallel starlight falls within the telescope aperture and behind the collimator, it occurs in parallel and compacted again. Here one could also install a simple unit such as the Staranalyser (transmission grid + camera + detector) and would thus produce a simple spectrograph available, which is able already to show the spectrum of a very faint star at low resolution.

Virtually no light loss (High Efficiency)

Lightweight and small
High resolution attainable
Linear dispersion function
Interchangeable gratings

In the past 10 years, the reflection grating enforced in the amateur field. Meanwhile even enter the high-resolution Echellespektrographs into the amateur scene. They have the great advantage that the entire wavelength spectrum is mapped 350 to 700 nm at a time, divided into about 30 to 50 "orders" = part of spectra, which can then be concatenated by the software to a long range spectrum. For examples, see the Baches and the eShel of shelyak.

However, it should be stressed here again that the measurement task defines which type of spectrograph the "right one".

The well-suited for self-construction and in many ways for the amateur valuable type of spectrograph is the classical grating spectrograph with a rotating reflection grating (see opposite). The camera is due to the high resolution (wide "fan"), only a portion of the optical wavelength range from (in the graphic the green light). By rotating the grating the desired spectral range can be set on the detector.
My first spectrograph was built exactly according to this model.

The angle between the two optical axes of the collimator and the camera (= total diffraction angle) plays an important role. It should be small, so that grids can be used with a high stroke count (high dispersion). A 90-degree design, as it is realized in the Dados spectrograph, yields a low resolution spectrograph (up to 900 lines / mm grating at maximum). Gratings with a higher dispersion are no longer usable. My first spectrograph had a total refraction angle of 45 °, which also had not been optimal. I was able to use a grating with 1200 lines / mm (R = 6000).

An important characteristic of a spectrograph is its dispersion and the closely related resolution R. The higher the dispersion the higher the resolution R and the more details can be discerned in the spectrum.
This resolution is in Staranalyser still low. But then one has the complete spectrum in the visible light ( "optical section") on the detector. Is due to higher dispersion the light diffraction range angle opened so far that he some colors falls outside the detector, we caught only a portion of the spectrum of visible light on the detector, however, higher resolution and more detail showing (as it were, magnified). We are seeing already from an essential criterion: the desired dispersion depends on the observation target. What else do you actually measure? The complete range of spectral survey (low resolution, R 1000) and classify the star or follow the profile of individual spectral lines studied (high resolution R> 10,000) or even the hyperfine structure (examined by lines at very high resolution R> 200,000)? Of these, the construction and the correct measurement function for the spectrograph depends on this goals. You have to decide your pricipal targets before you design and built a spectrograph, or alternatively buy. And the spectrograph must be adapted to the existing telescope (focal ratio) and also the CCD (pixel size and size of CCD chip)!!