Vanguard cards are special over-sized Magic cards (3" x 5" as opposed to 2 1/2" x 3 1/2") that offer a different way to play. Each card has an instruction that acts like a personal global enchantment, plus modifications to your starting/maximum hand size (in the left-hand circle) and life total (on the right). Obviously, both players have to agree to use them before the game begins. Below is the Karn card (placed next to the Karn, Silver Golem card for comparison).
These cards were used in several Arena seasons (starting in 1997), and were later available as part of a gift box. If you look around, you can find them at card shops and for sale online today.
Popular variants include each player randomly choosing a Vanguard card to play, or rotating Vanguard cards at determined intervals during a multiplayer game. But, of course, you can use Vanguard cards any way you want.
Here are the "official" rules, followed by the other seven cards from the first Vanguard series:
The Vanguard™ supplements consist of oversized cards that modify the game. A Vanguard card is selected before the game begins, adjusting a player's starting and maximum hand size and starting life total. Any abilities printed on a Vanguard card are played exactly like those of an in-play Magic card; however, these abilities have no color, and damage from them isn't damage from a permanent of any type or a source of any color. A Vanguard card isn't a Magic card, so it can't be affected by spells or abilities.