Windows Azure is an open and flexible cloud platform that enables you to quickly build, deploy and manage applications across a global network of Microsoft-managed data centers. You can build applications using any language, tool or framework. And you can integrate your public cloud applications with your existing IT environment. There are few options for you to start building your apps on this cloud platform.

Option 1:  3 Months FREE Trial
The FREE trial for 3 months is for any new customer. Each month, the FREE trial will allow you to use:

  • Compute: 750 hours of a Small Compute Instance* (can run one small instance full-time or other sizes at their equivalent ratios)
  • Storage: 20GB with 50k Storage transactions
  • Data Transfers: 20GB outbound / Unlimited inbound data transfer
  • Relational Database: 1GB Web Edition SQL Azure database
  • Access Control: 100k transactions
  • Service Bus: Free through March 31, 2012
  • Caching: 128MB cache

You will required to provide your credit card or debit card for identification, but you are capped by default at the free level to ensure you will never be charged.
To find out more details on how its work, you can click here

Option 2: FREE for MSDN Subscriber
If you are an existing Visual Studio Professional, Premium or Ultimate with MSDN subscriber, you get free access to Windows Azure each month, and up to $3,700.00 in annual Windows Azure benefits at no charge.
This offer provides a base level of Compute, Storage, Content Delivery Network, SQL Azure database, Access Control, Service Bus and Caching each month at no charge. All these offers is depends on your MSDN subscription level, the following is the monthly benefits table (please always refer back to WindowsAzure.com for the latest details):


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Yet to buy any MSDN subscription and interested to buy now? Click here to find out more details.
http://msdn.microsoft.com/subscriptions/hh442902

Option 3: Purchase Window Azure
There are 2 options for you to purchase Windows Azure:

1) Pay-As-You-Go
This pricing option is extremely flexible. It involves no up-front costs, and no long term commitment. You pay only for the resources that you use.
In the Pay-As-You-Go option, Web, Worker, and VM compute resources are paid for on a per-hour usage basis. Storage, Database, Bandwidth, Caching and CDN features are charged on a per-GB/month usage basis, with per-transaction costs for some resources.

2) 6 Month Plans (save up to 39%)
Buy a 6 month plan and take advantage of significant price discounts off our Pay-As-You-Go rates. You will get a 20% discount for Compute and Database and up to 39% off for Storage. The storage discount depends on the number of terabytes that you purchase.
Additional resource usage above the commitment level will be charged at the Pay-As-You-Go rates.

To learn more about paid Azure, click here and get an estimated cost from the calculator here.

Image: Paid Azure Pricing Calculator

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To explore more on what you can do with Windows Azure, you can always go to Developer Center.
http://www.windowsazure.com/en-us/develop/overview/