Connect to Live
First of all you need to subscribe to Xbox Live by creating your account. After this, simply connect to Live and visit the Marketplace.
In the Live Marketplace, you need to search for the games you desire. You can get full-game demos or arcade games and avail many other features. You can get demos of popular and most wanted games such as Call of Duty and Medal of Honour etc. Some of the arcade games which are available in Live are Zuma, Hexis and Bejeweled.
Choose the game
Next you need to choose the game which you desire. You can always download the free demo version before buying the whole game. Arcade games mostly come with a free demo version and a paid full game.
Use the Microsoft points which are an online version of cash in order to buy the games. These points can also be used for renting movies, getting themes and several other things besides downloading a demo.
Get the Microsoft Points card
You need to get a Microsoft Points card in order to buy demos and games form the Live Marketplace. After getting the card, simply go to Redeem Card option in order to buy things. These Microsoft Points cards are easily available from any local gaming stores.
Microsoft Points online
Instead of buying a Microsoft Point card, you can get the points online. Use a credit card to purchase Microsoft Points. Remember to ask your parents beforehand if you are using their credit card.
Get the full version from store
You can always buy full version from any local gaming store. Even if you do not buy the game, you will have many games or demo version of games in your Xbox 360.