Current ISO standards for rating film sensitivity measure the exposure needed to achieve a particular level of exposure in the deep shadow areas of an image. This measurement is relatively insensitive to the development time used.
However in the mid-tones and highlights, extending development beyond the standard duration causes a significant increase in density. Since the mid-tones of a photo are generally where the most important detail resides, it is possible to "fake" an increase in film sensitivity by deliberately underexposing, then compensating with extended development.
This is known as push processing. Understandably there is always a loss of shadow detail (pushing does not increase this), and an overall increase in contrast. The graininess of the image will also worsen. However aesthetically, these effects may still suit the image—for example in gritty reportage photography.
It is incorrect to say that push processing "increases" a film's ISO. The ISO sensitivity rating is defined under certain standard conditions, and is fixed when a film is manufactured. What push processing accomplishes is to allow a higher practical working sensitivity, and numerically this number is called the film's E.I. (exposure index).
In black & white processing, an old rule of thumb was to increase development time by 30% to gain one stop of speed (a doubling of the E.I.).
If you study the spec sheets for Kodak's T-Max 3200 and Ilford's Delta 3200, you will notice that 3200 is quoted only as an E.I. Under the ISO rating system, both films have a sensitivity of approximately 1000. However they have been engineered so that push processing to an E.I. of 3200 will give an acceptable level of contrast and grain for many purposes.