What a wide open question, and replies show it.
Three reasons for shooting film at a different ASA setting than stated on box.
1/ To Push/Pull process.
Ie; dont have fast film to hand; shoot 400 rated at 1600 and Push-Process' give it longer in the tank to compensate. Comes out grainy and contrasty, but you get shots. Alternatively; don't have slow film to hand, bung in 400 and expose at 100, and shorten the dev-time.
Sometimes done for convenience, sometimes for effect, some-times for cost.
Some films also designed for push/pull processing and may come with dev compensation times.
Used to do this a lot with bulk-loaded Croatian Agfa copy stock in B&W and Slide. Got a 30m roll of ASA400; roll into cans, and attach masking tape lable, stating 'B&W' or 'SLIDE' then when I shot it, would mark what ASA I rated it at before putting it in the camera. Would use it as either 200, 400, 800 or 1600, to save buying lots of different rated films.
2/ Frame By Frame Exposure Compensation
My old Olympus XA2 compact has no exposure compensation dial. Want to add 2/3 of a stop over exposure to compensate for back-light? Do it on the ASA dial. Same with the OM10, and my Konica C35, which are all meter-coupled AE exposure. Its the only way to get any 'manual' over-ride to meter suggested settings.
3/ Base-line adjustment
Film or circumstance dependent; 'correction' to metering.
Eg: If you know your meter has tendancy to over-expose, compensate by upping the ASA setting, to 're-calibrate' without taking it to bits.
Shooting slide film; was always advised to under-expose by aprox 1/3 stop, to avoid blown highlights and saturate the transparency for projection. So if I rated a slide film at 400 'box-speed' I'd then compensate one click under.. no idea what that would be, Olympus just have two inter divisions between major ASA settings!
Other-wise; shooting a set; rather than adjusting frame by frame, as all are going to need much the same adjustment; say when skiing, and everything's white & meter will tend to under-expose, presuming 18% grey; or at a Rock-Gig in dingy student gym, where dark, and meter will presume black stage is lighter and try and over-expose.
So basically; for various reasons, one way or another, very seldom set box ASA speed and been a slave to the meter and its couples exposure settings.