There are various methods for measuring protein concentration, the most common ones are listed below.
Absorption at 280 nm
Measuring the absorption at 280 nm is a simple method and can be done for clear protein solutions without components that show absorbance at this wavelength (e.g. nucleic acids, chromophores, detergents). For calculating the extinction coefficient, you can use online tools like Pepstats from the EMBOSS package or ProtParam on the Expasy website.
The following list of assays make use of the binding or formation of a chromphore in the presence of soluble protein and measuring the absorbance of this chromophore. Each of these classical assays have advantages and disadvantages regarding accuracy, robustness or compatibility with various buffer components.
A modified version of the Ninhydrin assay described by B. Starcher (Anal Biochem., 2001, 292(1), p125-9, PMID: 11319826) can be used for quantitative amino acid analysis after total hydrolysis of the protein.
Although this method takes longer than the others (includes total hydrolysis of samples over night and drying in Speed-vac), the hands-on time is relatively short. This assay is very accurate, robust and compatible with insoluble material like tissue samples or inclusion bodies.
There are also some newer reagents that show fluorescence upon binding to or reacting with proteins or peptides:
NanoOrange, CBQCA reagent, QuantITTM, EZQ (Molecular Probes/Invitrogen)
More information and a detailed comparison of different reagents can be found on the website of the Protein Expression Facility of the Wolfson Centre for Applied Structural Biology (Hebrew University of Jerusalem).