Choosing the artwork can be difficult enough! You've trawled all the galleries, art shows and fairs, and finally agreed on that piece you both really like. No compromises.
Think you're out of the woods? Well, you might be.
Option One: No Frame Required
Most works on canvas or panel are finished with a varnish (or the newest trend: resin) and do not need a frame. Usually they can be hung using a wire on the back, or even just on the actual wood 'stretcher bar' (that's the 'skeleton' of the canvas). No frame at all exposes the edges, and all that the artist has done to make the piece, which is interesting to see. It becomes part of the art in essence. Underpainting colour choices, splatter, and messiness all add to its charm.
If you like a more finished look, or if the artist was exceedingly messy with the edges, then by all means, frame it. Tip: keep it simple using a gallery floater frame as it is the cleanest look and suits both contemporary or traditional artwork. These frames usually come in black, gold or silver and will have a 'reveal' which is a space that separates the canvas edge from the frame by a 1/4" or so. Very chic and tailored as the art floats with that space around it which defines the canvas in an understated way.
Canvas works can get away without a frame or glass because they are protected from the environment by using varnishes or resins which 'seal' the paint used to make the art. A wipe down with a dry cloth gets rid of dust, and you're all set to hang you piece.
Paper works aren't so lucky.
Be it a photograph, silkscreen or etching, all will need a proper frame and glass in order to protect it. It will also need acid-free backing or mounting, and perhaps a mat on top if it so suits the artwork in question.
Just Shadowbox It
The easiest and neatest look for a work on paper is shadowbox style framing. It incorporates the said necessities above but the artwork is 'tipped on' to a back mat, thereby exposing any of the paper's lovely deckled/fringed edges, and set back from the glass using an inner wall, so the piece appears to be floating in a box. Anything can be shadowboxed and will always work out well for you. If you'd like to frame up that medal you won in primary school or maybe your grandmother's needlepoint, or anything three dimensional, a shadowbox is the way to go.
I Would Mat It
Photographs are usually printed on photo paper which usually doesn't warrant a shadowbox style frame. The nature of photography naturally lends itself to mounting the artwork (although some schools of thought say otherwise, I like to mount photographs as they need to have a flawless flat surface, and without mounting you can get unwanted 'buckling' of the paper). It also looks great when the artwork has a nice big white border. A large white mat can create a visual break for the eye, and causes the viewer to focus in on the jewel in the middle, that is, the art.
When matting any paper work always make sure the proportions are slightly bigger on the bottom. So, if you are showing 3" of matting on the top and sides, beef up the bottom to 3 1/2". Why? because if you do not, then the piece will appear to be sinking. Call it a trick of the trade, or optical illusion, but it's true, so make sure your framer does it.
Any paper work, be it a shadowboxed etching, or an Ansel Adams photo, should have some form of glass protecting it. There are many different types of glass out there these days. Regular 3mm glass is still my favourite but if you are framing something of great value either monetarily or just sentimentally, then high UV rated glass (or museum glass) should be used to protect it from any sun damage. Think of it as your artwork's SPF.
Lastly, now that you are well versed in how you may wish to frame your new art piece, as the boy scouts will say "Be Prepared." Good quality custom framing is always pricey. You might exclaim "but we didn't even spend that much on the art!", but that is not of interest to your framer. It's quite labour intensive to custom frame and can get up there in cost once you choose the moulding and glass (acid free matting backing and on top, mounting, and labour all are added up in that deal too).
Remember that a great frame can either make the piece of art come to life, or it can overtake the piece, so be careful. You want to bring out the best in the artwork. You want your friend to say "Wow that artwork is amazing," not "great frame!" (and certainly not "oh it goes with your room!"). So be wary of custom framers with high hopes to sell you that 5" Italian lacquered number that looks like a piece of furniture.
Most of all, don't stress out about it. Art is meant to be enjoyed -- framed or unframed.