A fast lens is one with a large aperture size. A larger aperture lets
more light through so the shutter speed can be faster (higher) - hence
'fast lens'.
The brightness of the image in the viewfinder is partly dependent on how
fast the lens is. The faster the lens, the brighter the viewfinder image.
Bright viewfinder images tend to make focusing easier.
I personally consider any lens with a maximum aperture of f2.8 or larger
(f2, f1.4, f1.2) to be a 'fast' lens.
The aperture 'f' number is roughly the ratio of the diameter of the
aperture to the focal length of the lens, so if you want to make a 200mm
lens with a maximum aperture of f2, the aperture has to be 100mm in
diameter. Since the aperture can not be smaller than the diameter of the
front element, the front element will have to be that size.
This is not meant to be a physics text book definition. With wide angle
lenses, one talks of 'apparent aperture' and the front element diameter to
focal length bit goes out the window.
A fast lens will therefore tend to be larger than a slow lens because of
the requirement for a larger front element. It will also tend to be
correspondingly more expensive.