First, you should know that geo replication is turned on for Azure STORAGE service, not for any other Azure services yet. Then, you shall also be aware that this geo-replication is for disaster recovery mainly (and only as of today).
If you have to replicate a DB (Windows Azure SQL Database, a.k.a. WASD) you can use the SQL Data Sync - the only known way as of today to sync Azure Databases (either between different geo regions, or between Azure and on-premises).
There is not support for Windows Azure Cloud Service geo-replication. If you need to geographically distribute your application, you have to manage cloud service deployment across different data centers on your own.
If this is the case, for Azure storage, I would suggest using a single Storage service for WRITE operations, but Azure CDN for READ operations. Otherwise it might get too complicated. Of course the chosen architectural approach will depend on the requirements of the app (and expected load).
Then, you have to combine the different deployments with Azure Traffic Manager with a "Performance" algorithm setup.
EDIT (NOV 2014)
As of Q3 2014, Azure SQL Database also support Geo Replication. And Azure Data Sync is depricated and removed service. Azure Storage replication continue to be offered with 3 different flavours: Zone redundant, Geo redundant, Geo redundant with Read Access.
And still no option to replicate between Geographic Regions (i.e. from EU to US). Replication is still only an option between Geo Zone pairs (same geography).
