Yes, you can use vintage/film Nikon F-mount (AI and later) lenses on a Nikon D3200, but you probably don't want to use adapters to use other mounts. Nikon F has one of the thickest registration distances of any SLR mount. This distance is how far the lens is held from the image plane, and needs to be maintained if the lens is to focus to infinity, as designed. Adapters add to that distance. This acts like macro extension tubes, and limits the far focusing capability. The mount throat diameter can be another issue, and is the reason Sony Alpha mount is not as adaptable as Canon EOS.
pre-AI lenses can be used on a D3200, but moving up to a D7000 or higher model, these lenses could cause issues, unless modifications are made. If you're willing to replace the lens mount, you can also use some Leica-R, Contax/Yashica, and Olympus OM mount lenses with a Leitax mount replacement kit. Otherwise, if an adapter exists, it will contain a glass element in it to act as a short teleconverter to get over the registration distance issue, and like any tc this decreases the maximum aperture, increases the focal length, and probably adds softness (especially if it's low-cost).
Realize also that with a D3x00/D40 or D5x00/D60 body, you will not have accurate metering. You will not have autofocus. You will not have lens information in your EXIF. There is more here than just popping it on and shooting. The phrase you probably want to google up is "non-CPU lens".
Canon EOS mount cannot use all vintage/film-era lenses, either. The six film SLR mounts you can easily find simple ring adapters for are Leica-R, Nikon F, Contax/Yashica, Olympus OM, Pentax K, and M42. Canon FD/FL and Minolta MD/MC are particularly problematic, as they have the thinner registration distance, and will require glass-element converters. Also realize that brand is not enough to distinguish a mount. Many brands had both SLR and rangefinder lines and any rangefinder lenses are out altogether, since they have even thinner registration distances than SLRs (think: no mirrorbox).
For example, if you look at the Canon Museum website, they list eight separate groups of lenses: EF, EF Cinema, New FD, FD, FL, R, S, and "special". Of those groups, only the EF ones are meant for dSLRs. FD/FL are manual-focus SLR. R was the pre-FL SLR bayonet mount, and S are (mostly) LTM/M39 rangefinder lenses. You need a collector's mindset to suss out all the possibilities of vintage lenses. There are a lot of exceptions and gotchas.
Because of the lack of electronic communication, there is no autofocus, no aperture control from the body (upshot: you can only use M and Av modes), and no lens EXIF information. There will, however, be accurate stop-down metering. Realize, too, that going wide is going to be harder (crop factor) with an entry-level body, and that the focus screens in dSLRS do not offer the manual-focusing aids film SLRs used to (split circle, prism collar). You can use an after-market focus screen, or Magic Lantern with liveview to get focus peaking and magnification, or learn to scale focus, but a modern, native-mount autofocusing lens will always be more convenient.
Two good sites with a lot of information and discussion about using adapted manual focus lenses are: