As it might be a bit tricky to get an accurate impression of the framed prints and the papers they are printed on from just the product descriptions, I will give my thoughts on the different paper types, and the specific papers I use, below.
Note that the pictures are intentionally darkened to show surface reflections and paper texture without blowing the highlights; thus they may appear bluish grey when actually white in colour. The general rule seem to be: the more blue in the picture and the darker the grey, the colder the white of the actual paper and the glossier its surface.
These tend to be well liked on account of their high contrast and highly saturated colours (and of course their general shiny glossiness). The downside of this is that they are be prone to show reflections in conditions with bright sources of light, and that they tend to attract fingerprints, cuts and blemishes rather easily. Because of this tendency to show reflections, I seldom use them for framed prints behind glass, as the glass adds its own reflections to those of the paper.
Tecco Iridium Silver Gloss 250gsm
This special paper shares characteristics with both the gloss and semigloss categories, but as its official name implies it is a glossy paper the gloss category is where I'll place it. The paper is stunningly beautiful for the right print, and it brings a metallic, "retro-futuristic", look to pictures. It can also add a mineral sheen to pictures of rock or a crystalline glitter to snow. As its reflections aren't as extreme as those of some other glossy papers, it tends to work rather well behind glass.
Maehlers Professional Glossy Photo Paper 260gsm
My most often used glossy paper. It allows for beautiful colour prints with deep shadows, but a tiny bit of colour information tends to be lost in these same dark areas. It has a relatively high weight and is therefore sturdy but without being stiff. It is more or less an equivalent to, and has the same strengths and weaknesses as, two other glossy papers I also occasionally use; Canon Plus Glossy II and the now discontinued Konica Minolta Glossy. Another paper with characteristics not unlike the Maehlers paper is Epson Ultra Glossy Photo Paper, which I use when I want to kick up the gloss into overdrive.
These are less populist, so to speak, as they give less of a wow factor and tend to show a narrower range of colours and not quite as deep shadows. Thus, they are more seldom used by hobbyists and they tend not to be the first choice when shooting family photos and the like. Despite this, they often have a certain something, and can give pictures a depth, presence and 3d feel not easily achieved with glossy papers. They are also favourites for Fine Art prints and art reproductions.
Epson Hot Press Natural 330gsm
A sturdy Fine Art paper from Epson which shows rich colours and deep blacks. It has a creamy, natural white colour and a smooth, but not entirely smooth, texture. I mostly use it for black and white prints, where its qualities are undisputed.
Aquarel Paper 200gsm
This noname art paper is not a photo paper and thus have a hard time showing as rich and vivid colours as papers with a dedicated inkjet layer. What it lacks in colour and contrast it makes up for in texture and artsy looks, however, and it does add a certain je-ne-sais-quoi to certain pictures. It works really well with black and white prints.
Mahlers Professional Photo Paper Matte 260g
This is the matte version of the glossy paper further up the page. It has a rather cold white look to it; good or bad is a matter of taste. The paper works well for many types of pictures and shows quite a bit of both saturated colours and deep blacks ... but it doesn't really shine in most cases. However, on some pictures, with large uniform areas of colour, the paper really shows its worth and adds a near-velvet look and feel.
Canvas Photo Paper 300gsm
This noname canvas photo paper allows for a wide range of both colours and shadows/highlights while also having a look and feel that simulates art canvas. This effect is somewhat spoiled by it being a paper (when all is said and done) and not a through-and-though canvas. This is evident in the sheer amount of detail the paper can hold; giving it an artsy texture but with a photographic real-life look to the print. I do also print on real art canvas, though, which tends to give fewer details to the print but with a more painterly quality to it.
Hahnemühle FineArt Papers
I sometimes use the terrific textured Fine Art papers from Hahnemühle in cases where I want to accentuate textures such as rock or wood, or when I want to strengthen the 3D-look of a photo ... or just because they are plain great. They do come in a myriad of looks and shapes, however; so you won't see any pictures of them here. The one I use the most is probably Hahnemühle Torchon 285gsm, however, as I love its rugged and accentuated texture with large dimples and ridges .
I would say that this is my favourite category of papers, which is reflected in the sheer amount of prints I've made with them. They tend to have descriptions such as "satin", "pearl" or "luster" to reflect their ability to show subtly beautiful highlights, glitter or shine but without outright reflections. As should be evident from the classification, semiglossy papers end up somewhere between the matte and glossy papers and have a mixture of their respective strengths and weaknesses (but to a lesser extent). Though they can be viewed as a compromise, many of the semiglossy papers have such a beautiful luster to them that the term compromise don't do them justice.
Epson Traditional Photo Paper 330gsm (Epson Exhibition Paper in the US)
A sturdy Fine Art photographic paper which tries (and succeeds) in emulating the look and feel of real analogue photo papers from yesteryear. It has a subtle but beautiful texture with an equally subtle sheen to it.
Satin Photo Paper 300gsm
Though a noname paper, this paper is one of my absolute favourites. It has a sheen or glitter to it that rivals the Tecco Irridium Silver Paper with the right picture. Looks beautiful behind glass.
Epson Proofing Paper Semimatte 256gsm
This paper is primarily intended for proofing and I therefore seldom use it for framed prints. However, it ain't half bad as a photo paper and it has seen occasional use.