Depending on the sophistication of the meter, it may display the capacitance only, or it may also measure a number of other parameters such as leakage, equivalent series resistance (ESR), and inductance.
Some checks can be made without a specialised instrument, particularly on aluminium electrolytic capacitors which tend to be of high capacitance and to be subject to poor leakage.
Ultimately, in difficult cases routine replacement of capacitors (relatively cheap components) is easier than arranging measurements of all relevant parameters in working conditions.
Through the use of Kelvin connections and other careful design techniques, these instruments can usually measure capacitors over a range from picofarads to farads.
Modern bridge instruments usually include a digital display and, where relevant, some sort of go/no go testing to allow simple automated use in a production environment.