Whether or not games can be considered art causes a lot of contention.
Article on Gamasutra
Games Are Art
Roger Ebert’s article

My opinion, obviously, is that they ARE, for the following reasons:
1. They can be judged as good or bad.
2. They can indirectly express to the user’s what is not expressible through direct explanation.
3. They can effect a user’s emotional state.
4. They can have subtext, as in more depth that appears on the surface.

(I realized I wrote the above definitions with “observer” initially, and I had to change all those words to “user”.)

One of the arguments against games being art, according to Roger Ebert and some others, is that they do not have complete authorial direction; the author does not have complete control over what happens. I guess it seems to him that every time a form of media becomes briefly interactive, then it ceases to be art and briefly becomes something else, becoming art again once the interaction phase is over. But, I want to ignore any arguments about that here and instead try to figure out what parts of a game (if its even okay to consider them separately) can contribute to the artistic value of the whole game.

Traditional Storytelling
Games can and do have excellent stories, comparable or better than anything seen in film, books or television. This type of art is well-known, and telling it through a game’s viewpoint is very interesting. Examples of this include The Dreamfall series and God of War, both of which are well-known for their story-telling ability. As games grow up, they begin to find unique ways to tell stories. The Half-Life series is famous for this, telling the entire story with no narrative explanation from the first-person viewpoint as events are happening.

Interactive Storytelling
This is a new form to gaming, essentially derived from Choose-Your-Own-Adventures, but much more flexible. Good examples include The Wing Commander series and Deus Ex. These both have branching storylines with multiple endings. There is a lot of academic work in this field, which I keep track of at a blog called Grand Text Auto.

Traditional Audiovisual Aesthetics
This is what is usually shown in the form of screenshots. I, personally, think that this work is very admirable but effort could be expended much better elsewhere. Still, there are some very beautiful games out there, such as Myst or Riven or Shadow of the Colossus. Unfortunately, when a game is advertised, this is usually the only part of the game that the consumer sees.

Formal Rule Sets
This is where games started long ago. Chess is a good example, so are modern RPGs, or the MMORPG Eve Online, which has over 30,000 players and a persistent virtual economy. Games that have too much emphasis on this part of their design tend to only be attractive to a small audience, something the predominantly science and engineering background male games industry struggles with. I’d also like to extend the idea of formal rule sets not just to slow-paced RPGs but also to faster-paced First-Person Shooters. When you are fighting in an FPS, your thinking can be quite advanced, for example:
“The enemy has a Covenant turrent placed behind the rock and there are three grunts with sticky plasma grenades. I can’t drive my Warthog in because they’ll stick me so maybe I’ll drive a Ghost over and ditch it while equipping a needler so as to take out the Gold Elite. I’ll save my only frag grenade to turn over the turret and take the dead Gold Elite’s plasma gun to take out the remaining grunts.”

Fertile Area for Emergent Gameplay
When a game has formal rule sets, but there are no clear goals, emergent gameplay occurs. This is a bit of a buzzword in the games industry right now. Anything by Will Wright is an excellent example of this (Sim City, Sim Life, Sim Earth, The Sims, upcoming Spore). These are really fun, and they have a very wide audience appeal.

The Action-Response Relationship with the User
This is the way in which the game (on whatever platform) responds to user input. Of all the above-mentioned experiences, this is unique to electronic interactive media, and what I’m really interested in. This is the same effect as improvised interactive theatre, where the user is playing a role. Through this method, games can do amazing things, like explore the nature of experience and sanity. Advanced interactive storytelling is also possible, such as with Facade. Examples of this are rare, but they are becoming more common. The best example I could find is a game called Eternal Darkness:Sanity’s Requiem where there is a “sanity meter” which depletes during gameplay. As the sanity of the player gets lower and lower, strange things happen in the game world, and the fourth wall is broken with terrifying effect.

The reason why I find games so attractive is that they have so many different ways to express themselves, and I think we are just hitting the tip of the iceberg.