I have a Nikon D3200 and I want to buy another lens for portrait photography. The lens I want to buy is an Olympus, not Nikkor. Would it fit my camera or would I need a lens adapter? And are there different types of lens adapters? If so, which do I need?
Unfortunately, no. I'm going to assume that the lens you are interested in is a current Micro 4/3rds Olympus lens, but this is true for vintage "OM" lenses as well. The Olympus cameras have a shorter flange distance — basically, the lenses mount closer to the camera, which is the reverse of the situation where an adapter can work. (It might be theoretically possible to use a lens with glass elements to correct for this, but the expense and image quality loss with make it not worthwhile.)
You should probably look for lenses with a native Nikon mount for best results. Or, since you have an entry-level Nikon camera, if you're really attracted to what Olympus offers, it might not be that big of a deal to switch systems — sell your D3200 and buy a new Olympus to match your desired lens.
If you are speaking of a micro four-thirds Olympus m.Zuiko or four-thirds or e.Zuiko lens, then no, you cannot use them. Not only is the registration distance (the distance from the sensor to the lens mount) much smaller than Nikon F mount (which means you could not achieve focus at infinity with the lens without an adapter with a glass element to act as a teleconverter, which will reduce maximum aperture, increase focal length, and probably introduce softness), they are designed to cover a smaller sensor, so they'll also vignette.
And vintage Olympus OM-mount (manual-focus film) lenses cannot be adapted to Nikon F with simple ring adapters--you have to replace the lens's mount to make the registration distance match. The only readily available adapters that I know of are the Leitax mount replacement kits, which don't work with all lenses--and at US$80-100 each (depending on the exchange rate today), they're not cheap. Understand, too, that this is essentially shooting with a non-CPU lens: there is no electronic communication between the lens and the camera body. With a D3200, this means you have no accurate metering. You have no lens EXIF information. You cannot shoot in modes other than A and M (because the body can't control the lens's aperture setting). And you will not have autofocus. This is not for the faint of heart, nor is it an "easy" way to save money. And don't forget that the crop factor is still in effect. Lenses that are designed to be "wide angle" on full-frame tend to be less-than-wide on a crop body.
This kind of lens adapting is much easier on the Canon side of the fence, because the Canon EOS-mount registration distance is smaller than most of the manual dSLR mounts, so simple adapter rings can be used. And unlike Nikon D3x00 or D5x00 bodies, all Canon dSLR bodies can perform accurate stop-down metering without any electronic lens/body communication. On the Nikon side of the fence, adapting manual lenses like this is much much more of a PITA.