Astronomical magnitude is an indicator of the brightness of a star (or other astronomical body) based upon differentiation of brightness by eye, which has been formalized as a logarithmic scale with lower numbers brighter, essentially -2.5 times the log 10 of the flux density. Some magnitudes (as viewed from Earth):
Relative magnitude of two objects:
m1-m2 = -2.5 log10 I1/I2
These are traditionally scaled so Vega has a magnitude of zero, though more recent methods take a specific flux density measure as zero. The above examples are apparent magnitudes, which is the magnitude as viewed from Earth, which is the typical meaning of magnitude if not qualified or in context. Absolute magnitude is defined as the apparent magnitude a star would have if it were 10 parsecs from Earth (the Sun's is 4.83). Bolometric magnitude is specifically a magnitude including all EMR wavelengths, i.e., including radio, infrared, etc. Magnitudes of stars are often cited for specific passbands or differences between passbands (color indices). With no such qualifications or context, it is best to assume magnitude means visual magnitude, an estimation of what you would see that uses the V band.
Magnitudes are cited for stars and other point sources, and also for extended sources, in which case they can be cited for the radiative flux from the source (e.g., as is cited above for the Sun, all the light from the object reaching us), or the source's surface brightness (of all or a portion of it), often used in describing galaxies.