Both games can be found among others here
http://www.wizards.com/
I can't get into details about the games but a short description for each could be:
Magic the Gathering is a game with collectible cards. You buy or trade cards building your deck. Each card has special abilities and you can combine abilities and form strategies in order to win your opponent/s. The game is addictive. However you need to spend money in order to build a decent deck. New cards are being released every three months, which you can ignore if you play for fun or pursuit if you want more abilities to choose from or you're into tournaments.
Dungeons and Dragons is a Role Playing Game like the ones you get on your pc or the internet. Then pen and paper form of dungeons and dragons require only a group of players, knowledge of the rules some dices and a GM or DM (Game Master- Dungeon Master). The person being the DM prepares ahead of the meeting, an "adventure", a tale with plot and precalculated ways to evolve based on the reactions and the decisions of the players. During the meeting what's going on is like a live fantasy movie. Players (the heroes of the adventure) play their roles in the story tha is being directed by the dm.
In the game players can go kill monsters search for treasures in dangerous dungeons or anything else they can think of. The DM 'role plays' the bad guys, give descriptions and disrections to players and generally is the ruler of them all
for example during a meeting this dialog could happen:
DM: - You are walking almost for 5 hours down the narrow path leading deeper into the forest. You're feeling like someone is watching you...
DM: - Tell me the order you are marching..
Player1 (Thief) - I, the great rogue Znibl, am as usual ahead of the rest searching for traps of anything out of the ordinary...
Player2 (Mage) - I am at the middle walking behind my good friend Aragorn
Player3(Ranger) - I am with my friend the old mage
Player4(Fighter) - I am last watching our backs
DM: - Znibl roll a 20 sided dice
Znible rolls a 19 (good thing- if roll was 13 or less possibly Znibl wouldn't have see anything but that's not the case so the DM tells him:)
DM: - Znibl, with your hightened senses you manage to discern a shadow moving to your right! What do you do?
Player1 - I warn the others: Hey guys guard yourselves, we might have company!!
DM: - The minute you shout you see an arrow passing over your shoulder.
DM: - Two orcs apear from nowhere! Guys roll to see who plays first.
...the fight begins..
etc
etc
(..that was the short description??? :p )