The numerical aperture (NA) is a number (mostly shown on the objective itself) describing the amount of light coming from the focus that the objective can collect, and is directly related to the final resolution in the Image Formation. The numerical aperture of an objective is defined as the refractive index of the Lens Immersion Medium n times the sine of the half-aperture angle α.
NA = n sin(α)
A lens can collect more light the bigger it is (that is the angular part of the above equation: α is the angle subtended at the focus by the lens radius).
(The refractive index appears in the definition as a scaling factor. In this way, the NA determines the Spatial Resolution also scaling the wavelength of the light.)
Because sin(α) ≤ 1.0, the NA is always lower than the medium refractive index. If a 1.4 NA oil immersion lens is used to image into a watery object Total Internal Reflection (TIR) will take place, effectively truncating the aperture (see Acquisition Pitfalls).
The Numerical Aperture can be estimated from the angle of the cone as seen from the Point Spread Function in xz or yz view, in compress display mode (to increase the contrast of the low valued regions and reduce the contrast of the high-valued regions). In the following example, if the Medium Refractive Index n is 1.5 the numerical aperture is NA = n sin(60) = 1.3.
See the FAQ entry How do you measure the Numerical Aperture (NA) from the experimental PSF? and/or search the FAQ for "numerical aperture".