Focal length is a number expressed in millimeters (mm) that indicates a lens’s field (angle) of view. A The number itself refers to the measured distance between the point where light converges (the nodal point) and the image plane (sensor) when a subject is in focus at infinity. In practical terms, a smaller or shorter focal length equals a wider angle of view, and a larger (longer) focal length is a narrower angle of view. For example, 16mm is considered a very wide angle focal length, while 400mm is very telephoto focal length.
Technical explanation of Focal Length
To most photographers, the term "focal length" simply refers to a number on a lens that indicates how "wide" or "telephoto" that lens is. The higher the number, the more telephoto the lens.
However, the term itself does correlate to a specified "length of focus", and to understand this number we have to understand how a lens itself works, both in theory and in practice.
A lens' purpose is to focus light very sharply at a specific point behind that lens, like a magnifying glass focusing sunlight on a piece of paper a short distance behind it.
Camera lenses, however, have multiple glass elements and an aperture iris, which are all used in conjunction to sharply project a scene onto the image sensor inside the camera.
Of course an astute photographer will point out, those blue lines sure don't look very wide-angle, and that lens is in fact an ultra-wide-angle zoom! Is that diagram wildly inaccurate?
Indeed, each of the glass elements in a modern lenses will bend light in one direction or another. There are two reasons for this: first, to create a flat plane of focus instead of a curved one, and second, to create enough empty space behind the rearmost lens element to fit a shutter, and in the case of SLR cameras, a mirror.
This next diagram is also inaccurate compared to the actual optical formula of the particular lens, however it depicts the general concept of a multi-element lens construction: Light may enter the first element of the lens at a very sharp angle, and "bend" to a more manageable angle as it travels through the lens. In other words, if you wanted to create an 11mm lens on a 35mm full-frame sensor, with just one piece of glass, then yes, the nodal point of that lens would need to be 11mm away from the surface of the sensor, which is impossible on many cameras, and impractical or at least sub-optimal on all cameras.
Focal Length and Angle of View
Therefore, optics are used to bend light in a way that moves the nodal point forward away from the sensor, making the focal length distance purely theoretical.
This is why, in modern photography, a listed focal length is merely used to indicate that lens' angle of view when focused at infinity. A 50mm lens has a 46.8 degree angle of view. A 24mm lens has an 84 degree angle of view, while a 200mm lens has a 12.3mm angle of view. The two numbers are always inversely proportional.
Measured focal length
Unfortunately, the listed focal length of a lens does not always perfectly correspond with an exact angle of view, at normal shooting distances. Many lenses change their angle of view slightly when they are focused closer than infinity, and some lenses change their angle of view by a very noticeable amount. Therefore, some photographers measure the "true" angle of view when focusing on a subject at the distances they normally shoot at, such as 5ft or 10ft from the lens.