How could a hobbyist astronomer determine apparent magnitude of a star? Apparent magnitude is a rather complex way to determine the brightness of a star. Quoting the introduction text from the linked to Wikipedia page:


  The apparent magnitude (m) of a celestial body is a measure of its
  brightness as seen by an observer on Earth, adjusted to the value it
  would have in the absence of the atmosphere. The brighter the object
  appears, the lower the value of its magnitude. Generally the visible
  spectrum (vmag) is used as a basis for the apparent magnitude, but
  other regions of the spectrum, such as the near-infrared J-band, are
  also used. In the visible spectrum Sirius is the brightest star in the
  night sky, whereas in the near-infrared J-band, Betelgeuse is the
  brightest.


While it is of course a useful measure also for the slightly more casual, non-scientific observer to help determine the observed star from its neighboring cluster, or identification in general, I always wondered if there is a way to measure apparent magnitude with enthusiast-class equipment, and what would these procedures be?

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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Apparent magnitude scale and observational limits (Source: ESA Science)

Also interesting would be a description of what level of precision and enthusiast can get with such equipment while measuring apparent magnitude of a distant star. If you need specifics to answer the question, such as precise available equipment or subject of observations, please feel free to choose at will any that would broadly match the capabilities of enthusiast-grade equipment.
The easiest way to determine the magnitude of a given star is probably to use the Pogson relation. The idea is to determine the magnitude of a star knowing the magnitude of a reference star; it is thus quite easy, using a well-known reference as Vega or Sirius.

The Pogson relation is given by:

$$m_1-m_2=-2.5\ log\ \left({\frac{E_1}{E_2}}\right)$$

where $m_1$ and $m_2$ are the magnitudes of star 1 and star 2 (your reference star), and $E_1$ and $E_2$ brightness (that could be in arbitrary units, which is a good news for you).

With a CCD detector the brightness is easy to determine since it's just the flux of a pixel. You don't have to convert it in physical units (as Watt per square meter) since you have a ratio $\left({\frac{E_1}{E_2}}\right)$, so the raw intensity is enough to determine the magnitude.

As for any other measurement with a CCD camera, you have of course to determine the bias, dark and flat-field and to reduce your data before estimating any magnitude. The precision will depend on the number of observations performed and, as always, it is better to perform several short exposure images than a single long exposure one.