Exposing and processing are two different things. You can expose normally, underexpose, or overexpose -- and process that normally. You can also pull or push process something normally exposed, overexposed, or underexposed.
Both exposure and processing affect DENSITY, the amount of silver (in b&w) or color dye that is formed.
The manufacturer's recommended exposure index (EI) basically means that if you expose correctly at this rating, and process normally, you should end up with a negative of average density.
But you could change the final density of the negative, which in turn affects the printer lights needed to create a print of normal brightness (or density, again). For example, underexposure leads to the "thin" negative (less dense) that requires lower printer lights to lighten the image back to normal. Overexposure leads to a "thick" or "dense" negative that requires higher printer light numbers to darken the image down to normal in the print. Here I'm talking about normal processing for this under or overexposed negative.
You could also underexpose (would cause a thin negative) and then push-process (extended development) to increase the density back to normal. This is what would happen if you underexposed by one stop and then asked for a one-stop push-process.
You could even overexpose by one stop and then push-process by one stop, ending up with a total of two stops extra density, just as if you had overexposed by two stops and developed normally.
So you see that exposure and processing are two different steps that both affect density. You can alter one, the other, or both.
How far you can over or underexpose, process normally, and then print up or down to restore the image to normal and find this acceptable-looking is a matter of taste. Generally people try to print in the middle of the printer light scale or slightly higher (i.e. expose for a slightly denser negative and then print down.) A thin negative printed up to normal tends to look milker & grainer than a thick negative printed down to normal, which tends to have deeper blacks and richer colors. But within reason, like no more than a stop of overexposure. I often rate a stock 2/3's of a stop slower than recommended and then print down, so I'd be rating a 500 ASA color negative stock at 320 ASA and usually am printing in the mid 30's instead of the high 20's on the printer light scale to get a print of "normal" brightness.
Push-processing tends to increase grain and contrast; pull-processing tends to decrease grain and contrast and also mutes the colors.
Going for a denser-than-normal negative is less necessary when you are shooting negative for scanning or telecine transfer. It mostly benefits negative for printing.