Microsoft Azure is a cloud computing platform and infrastructure, created by Microsoft, for building, deploying and managing applications and services through a global network of Microsoft-managed and Microsoft partner-hosted datacenters.  Included in this platform are multiple ways of storing data.  Below I will give a brief overview of each so you can get a feel for the best use case for each, with links provided that go into more detail:

• Local Storage: Provides temporary storage for a running application instance.  Local storage represents a directory on the physical file system of the underlying hardware that an application instance is running on, and can be used to store any information that is specific to the local running application instance.  You can create a number of local stores for each instance
• Windows Azure Storage:
  • Blob: A reliable, cost-effective cloud storage for large amounts of unstructured data, such as documents and media files.  It is a highly scalable, REST-based cloud object store.  The storage service offers three types of blobs: block blobs, page blobs, and append blobs.  Block blobs are best for sequential file I/O, page blobs are best for random-write pattern data, and append blobs are optimized for append operations
  • Data Lake Store: This is “HDFS as a service”.  It is a new flavor of Azure Blob Storage which can handle streaming data (low latency, high volume, short updates), is geo-distributed, data-locality aware and allows individual files to be sized at petabyte scale.  See Azure Data Lake
  • File: Fully managed file shares that use the standard SMB 3.0 protocol.  Migrate file share-based applications to the cloud with no code changes, allowing you to lift and shift legacy apps
  • Disks: Persistent block storage for Azure Iaas VMs.  The disks are actually .vhd files that need to reside in an Azure storage account.  You have the option of using Premium storage disks: SSD based, high IOPS, low latency
  • Table: A NoSQL key-value store for rapid development using massive semi-structured datasets.  Highly scalable to PBs, and has dynamic scaling based on load.  Has fast key/value lookups
  • Queue: A way to reliably store and send messages for scenarios including workflow processing or communication between application components.  Can decouple and scale components.  Helps your applications absorb unexpected traffic bursts and can prevent servers from being overwhelmed by a sudden flood of requests.  Instead of getting dropped, incoming requests are buffered in the queue until servers catch up—so traffic bursts don’t take down your applications
• SQL Database: Fully managed database-as-a-service that is in the cloud and is based on the Microsoft SQL Server engine.  SQL Database delivers predictable performance, scalability with no downtime, business continuity and data protection—all with near-zero administration
• DocumentDB: A NoSQL document database service that natively supports JSON and JavaScript directly inside the database engine. It’s a solution for applications that run in the cloud when predictable throughput, low latency, and flexible query are key

More info:

Data Storage Offerings on the Azure Platform

Azure Table Storage and Microsoft Azure SQL Database – Compared and Contrasted

Introduction to Microsoft Azure Storage