For instance, the chances that a player owns a Black Lotus are slim, but if s/he has his/her opponent's permission, the player and opponent could play with the understanding that a different card represents a Black Lotus. A proxy may be a card with different text written over it, with a piece of paper with the text written on it taped over it, with the text paper inserted into its card protector over it, or anything else the players agree upon.
Another use of proxy cards is to protect high value cards. For instance when you do actually own a Black Lotus or perhaps a foil Jace, The Mind Sculptor you could put a proxy card of it in your deck while having the actual card in a special protector on the side, to prove that you actually own the card. However when no proxy cards in the format you're playing be ready to have your deck inspected to check if proxied cards equal the authentic cards you own.
The website magiccards.info allows you to easily print copies of cards as proxies.