In general, if you increase development, you increase contrast: the linear part of the H&D curve (log exposure vs. density) gets steeper. This generally has only a small effect on shadow details -- used in defining film speed -- but will push what would have been midtones to or near highlight density values.
This is commonly done for two reasons: one, if the film was (intentionally or not) underexposed; this can bring low values into a midtone, or two, if the scene was of very low contrast (gray on gray on a cloudy day, so no strong highlights or shadows). The latter case is the "legitimate" one; the former the one used by thousands of photojournalists and street photographers over the past century or more.
In either case, regardless of the exposure given, the separation between bright and dark tones will be increased -- as others have noted, this may result in highlights becoming unprintable or difficult to scan (especially if the film received normal exposure with a normal brightness range). Use of a compensating developer (like D-23, Pyrocat, or high dilution low agitation Rodinal) can help prevent highlights from blowing out, but will also reduce the level of contrast increase.
In my own darkroom, I use a Rodinal clone at 1:50 dilution, develop as if Push +2, but agitate only every third minute instead of every 30 or 60 seconds; I use this method to increase shadow detail (by giving maximum development) without much contrast increase (because the developer will exhaust locally in high exposure areas between agitation cycles).