For instance, the chances that a player owns a Black Lotus are slim, but if they have their 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 are allowed in the format you're playing, be ready to have your deck inspected to check if proxied cards equal the authentic cards you own.
Wizards of the Coast has no desire to police playtest cards made for personal, non-commercial use, even if that usage takes place in a store. However, cards used in DCI-sanctioned events must be authentic Magic cards.
In sanctioned tournament like Grand Prixes, if a certain card(s) suffered a damage beyond suitable for competition, head judge of the event has the right to issue proxy card(s) replacing damaged card in deck for that event. Players must also keep the respective damaged card(s) and show to the opponent if necessary(like setting aside when the card comes into play) if a proxy cards replaced damaged cards.
Wizards remains committed to vigorously protecting the Magic community from counterfeiters.
The websites like Scryfall allow you to easily print copies of cards as proxies.