When you host your application Azure you don't get SMTP server or any other mail services.
So you got 2 choices. 1) Host your own SMTP server or 2) Use SendGrid where you can get 25000 emails/month for free.
Start by registering for SendGrid in this url.
Fill in the form and create your account. (Doesn't require credit card).
It will take them a while to provision you an account. If you have any problems with provisioning don't hesitate to send email to [email protected] They are very fast to fix your problem.
While it's provisioning create a new WCF application. Remember to choose .net 4.0
Then search for SendGrid and install it.
Now edit IService1.cs and Service1.svc
Now you can send emails easily from your Azure application.