All of the information I will be using comes from Wikipedia: Proxy Card.
People who make proxy cards either do it by hand (rare, but more accepted, especially as an altered card) or using a card maker program. Those that use the programs use whatever artwork they may happen to like, and rarely credit properly. As the proxy card is made using a template which is copywritten to the owners of the game there are a lot of issues there. (One of the Magic: the Gathering proxy card design programs was removed because the creators were sued by WoTC.)
Now on to the information that was promised about Proxy Cards.
Common use of proxies
Proxy cards can be used in various situations. The rules and restrictions are object of common agreement, or a given policy, and may differ from the above mentioned "fair play requirements".
* In casual games, the players may agree on a common policy of how to deal with proxy cards. This allows to play a higher variation of card combinations and strategies, while keeping a limit on the expenses.
* In tournaments, the organiser may permit a limited number of proxy cards, and define rules about how these cards must look. This policy has become especially popular in games or formats where some vital cards are far too expensive, such as the vintage format in Magic: The Gathering.
* For playtesting. Proxy cards allow a player to test new cards, before he or she decides to actually buy or trade for them.
* In card prototyping. Card developers in companies like Wizards of the Coast use proxies to playtest their ideas for new cards before they are printed.
* Some players create cards based on their own ideas for card themes and mechanics. In this case, however, the term "proxy" may no longer be applicable, as these cannot be considered substitutes for existing objects.
* An extreme form of copying original cards is forgery of expensive cards, which is again outside the "proxy" category.
Famous cards that are often proxied are the so-called power nine in Magic: The Gathering, which are considered totally out of balance in gameplay, while being unaffordable for the average player, due to their rarity and enormous price on the secondary market.
A friend of mine (actually a supporter of my dA account) makes M:tG proxies and art for M:tG proxies, you can find all of this at: http://mtg3dfree.end-day.net. She doesn't use them in play, and has only considered selling copies of them as poster prints to raise money to get more resources for the programs she uses. She actually has the images licensed and everything.
As far as I'm concerned these proxies are made mostly using artwork that the user cannot or will not credit and therein are stolen. What my friend does is a free service, she lets them use this artwork.
A proxy card really is not allowed in play, but there are always exceptions to that rule.
Rules when playing with proxy cards
Players who do not trust each other blindly will usually agree on common rules, defining which kind and number of proxies, if any, is allowed for each player. These rules may vary drastically depending on the participants and the occasion. However, some restrictions are implied naturally by common sense and plain physics.
 Indistinguishable on the Back
The main issue to guarantee fair play in a card game is that all cards in the deck must be indistinguishable for any player who does not view the front side (if card sleeves are used, the term 'card' means the sleeve with the card inside). Ideally, all cards should be indistinguishable in the following characteristics to effectively prevent cheating.
* Card size and shape, including the typical rounding cut on the edges.
* The card's total weight, its center of gravity and, ideally, the moment of inertia (which implies a homogeneous distribution of mass on the surface).
* Overall and local stiffness and elasticity - all cards should behave equally on bending.
* Overall and local thickness.
* Feel and relief (tactile characteristics) of the card, especially elevations and cavities on the surface on both sides.
* The image printed on the back side, including its shininess.
* Overall and local transparency, when examined with a light from behind.
Besides these physical implications, it should be considered that someone (the players or a judge) will need to control the validity of the cards - which may prove difficult with some of the above points.
Therefore, the use of proxies is sometimes further constrained to only one method of fabrication, for instance.
The difficulties of control can also be an argument to totally prohibit the use of proxy cards.
Unambiguous Mappings on the Front
Once the front of a proxy is revealed to the other players, it must be clear to everyone what it is meant to substitute. The decisions of what a player's proxies are meant to substitute must be made before starting play. If two proxies are meant to substitute different cards, they must be easily distinguishable by looking at their front side. Ideally, the label of a proxy should be enough to tell what it's meant to substitute. Alternatively, a legend or agreement can be used to prohibit players from changing these mappings during play.
Another issue for the front side labeling is to maintain a fluid gameplay. Poor labeling will likely cause unpleasant disruptions, even slips and mistakes caused by accidental confusion. It is therefore desirable that each proxy is labeled with the name of the card it substitutes, and its basic game-relevant characteristics, and erase all decorations and printed information that may be misleading. Any relevant information that is not written on the card should at least be found in the legend.
 Agreements and coded rules
Additional rules can restrict the number of proxy cards allowed in a deck, and the physical ways in which they may be created.
Such rules can be a simple agreement between two players, or they may be defined by the host of a tournament.
"Official" Magic: The Gathering tournaments sanctioned by the DCI do not allow the use of proxy cards. Tournaments hosted by game shops or other third party organisations are not bound to that policy, however. As mentioned above, in more and more such tournaments, participants are permitted to include 5 or even more proxy cards in their decks, at least when playing the cost-intensive vintage format.
Still, I have sent in an inquiry about deviantART's stand of posting proxy cards to user accounts. I will update this entry when I have received an answer (give me about 72 hours). Let's hope this will increase our knowledge about what a proxy card is, and why they are really looked down upon in the art community.
Edit(08/19/08): deviantART does not permit proxy card uploads. Have fun reporting!