From Diablo Wiki
Graphics is a term that refers to a game's visual components. It's not a technical term, but is more of a colloquialism; fans might say, "That game has awesome graphics." Or, as if often the case with detractors of Diablo III, "Ugly graphics."
Blizzard's Graphics Philosophy
Diablo 2's graphics were 2D, sprite-based, and the title was Blizzard's last game to feature non-3D graphics. Diablo 3 is being produced in a new, proprietary engine, and is fully 3D, as is virtually every computer game these days.
Blizzard games, as a company policy, do not attempt to feature the best, newest, highest-performance, most cutting-edge graphics. The company favors stylized imagery and bold outlines and colors over photo-realism, and gameplay over any visuals. Blizzard also works to keep their products playable on sub-top of the line machines, thus widening their potential consumer base. It's been a successful strategy thus far, and is one they'll continue with Diablo 3.
Many MMORPGs released since World of Warcraft have had objectively "better" graphics, but fans don't buy games because every blade of grass is rendered individually. They buy them because they have a nice look and play well, and WoW's graphics, though years old and never that technically advanced to begin with, remain popular and visually pleasing to most gamers.
Bashiok explained their company philosophy towards this issue in a forum post from December 2011.
Realism vs. Style
One of Blizzard's design goals is to create distinctive outlines and strong color schemes. Their goal is not to push the highest polygon counts or to make their games look the most "realistic." This is easily seen in the graphics for their games, which are always visually-pleasing and colorful, but can definitely be described as "cartoonish" or "stylized." Again, this is the goal; none of the characters or monsters in World of Warcraft look like real people, and they're not meant to. Blizzard's art style can be analogized to how people look in a Pixar film; recognizable, but exaggerated and stylized, rather than realistic.
This approach is in part responsible for the lasting popularity of most Blizzard titles. Millions of fans continue playing five and ten year old Blizzard games, while new games come and go. Gameplay has a lot to do with this, but graphics matter as well. Realistic graphics look amazing when they are new, but after a year or two there are always games with much better effects, making last year's state of the art look dated. Blizzard has never chased that brass ring, intentionally striving for style over higher polygon counts.
This approach pleases most fans, but there are always some people who equate "realism" with "better graphics," and who look at WoW or D3 and complain that they have "ugly graphics." What such people seem to mean is, "not realistic enough," since the comparison is always to some new game with a higher polygon count.
That's a valid point of view, but it's not really an argument; tastes vary, and it's not as if Blizzard is trying to make Diablo III look like Crysis (or whatever the next game with super-realistic graphics will be). Diablo III looks how Blizzard wants it to look, with an art style influenced by their design goals, and by a design philosophy that the game should be playable at a good frame rate on less than state-of-the-art computers. This is therefore a fairly pointless and frustrating argument, since non-fans are attacking the game for not being something it wasn't designed to be.
In September 2010 Diii.net posted an article that tried to bridge this divide with an analogy about the realism of works of art. Everyone has different taste in artwork, but hardly anyone judges paintings by which ones look most like real life. If they did then everyone would prefer photographs to paintings, and while some people do like photos best, most people like art, of all different types, more than photos. It's unclear if the people who say that Diablo III has "ugly graphics" have some essential difference in their visual preferences that makes them dislike most art while always preferring photography, but it's not an impossible analogy.
Too Many Special Effects?
Though Diablo 3 is not being designed to be the shiniest, most graphically-amazing game ever produced, it does have a lot of nice visual effects. The team wants it to be pretty, but the visuals must support the gameplay. There's been some debate about just how many explosions and visual starbursts there should be.
- Diablo-Source: What are your thoughts on the special effects and death animations since all the feedback fans gave at Blizzcon?
- Julian Love: Well, actually, the feedback I found that the fans gave mirrored our own internal feedback. It turns out that in order to get the Monk and a number of other features for Blizzcon we just slapped a lot things into a special effects kit that were not meant for how often they showed up in the game and made the Blizzcon build a little messier than we had intended it to be. However, the way we tend to go about doing this kind of thing is to make everything too big on purpose to make sure that we've gone far enough. Once we realize that we've done that, we go through a period where we sort of pull things back. It's kind of akin to mixing music where you sort of record everything in full volume and then you go to the mixer board and you tweak knobs until you tighten everything up and get back into reality. That's the mode we're in right now, tightening some things up.
Low-Res Character Models
One of the common "ugly graphics" complaints is that the character models aren't very high resolution. These are only seen in the character selection screen in-game, as well as on the official website, but it's a complaint that comes up fairly often. Bashiok addressed it in a forum post in August 2010. 
Which might actually be cool to do, we would love for the close-up characters on the website to look better than they do, but that’s allocating resources to what is essentially throw away material. So we haven’t deemed it worth the effort.
In general we also want to keep graphic requirements low so that a wide audience can enjoy the game, and also so we can have a lot of enemies on screen.
...We have scaling options too.Ultimately I recommend you stop looking at the zoomed in version on the website and instead look at movies and screenshots of the game play.
Most fans seem to understand this philosophy, since as the developers point out, Diablo III is not a game with a first person view point. The view of the characters on the website is almost never seen in the game.
See the Art Controversy article for more details on the look and art design of Diablo 3, and the huge controversy that sprung up based on the initial art design of the game.
See every Diablo 3 screenshot and artwork in the Image Gallery.