The breadth of services that Azure offers is pretty overwhelming, so take a deep breath :)
Cloud Compute
Azure Cloud Services allows you to create a compute service (ASP.NET, Python, node and PHP are all supported). This might be a worker role (e.g. a background tasks) or a web role (e.g. simple requests to display data). Cloud Services gives you the ability to scale these horizontally as needed.
Azure Batch runs your large scale parallel tasks and big batch processing jobs. Azure scales these as needed.
Azure Service Fabric is an orchestration layer for micro-service based deployments. It was borne out of internal use at Microsoft used to develop Azure itself.
Azure RemoteApp is a bridging technology (similar to Citrix type things) allowing you to access your application anywhere.
Azure Service Fabric is an orchestration layer for micro-service based deployments. It was borne out of internal use at Microsoft used to develop Azure itself.
Azure RemoteApp is a bridging technology (similar to Citrix type things) allowing you to access your application anywhere.
Web and Mobile
Web App Service allows you to deploy web applications in languages like C#, node.js and Python. It's now part of the more general Azure App Service.
API App Service allows you to deploy secured API services and generate appropriate clients to access them.
API Management Service lets you "take any API and publish a service in minutes". More specifically you get monitoring, RESTful and JSON-ful support and the ability to combine multiple back-ends into a single API endpoint.
Mobile App Service gives you an API specifically for mobiles, including support for off-line sync.
Logic App Service lets you integrate business processes and workflows visually. It's goal is to make it easy for you to join your data from on-premise to cloud-based workflows.
Notification Hubs provide scalable push-notifications to all major platforms (including iOS and Android).
Data
SQL Database provides a fully managed PaaS version of SQL Server with advanced features such as an index advisor (monitoring your workload to see access patterns that would benefit from an index).
SQL Data Warehouse is a data warehouse that can scale to huge volumes of data (pricing is based separately on Compute and Storage use).
Redis Cache is the PaaS version of Redis, an in-memory data structure store.
DocumentDB is a store for JSON documents. As of Build 2016, it was announced that there is a MongoDB compatibility layer (see here).
Azure Search provides a fully managed search service (with Lucene query compatibility).
SQL Data Warehouse is a data warehouse that can scale to huge volumes of data (pricing is based separately on Compute and Storage use).
Redis Cache is the PaaS version of Redis, an in-memory data structure store.
DocumentDB is a store for JSON documents. As of Build 2016, it was announced that there is a MongoDB compatibility layer (see here).
Azure Search provides a fully managed search service (with Lucene query compatibility).
Azure Table Storage provides you with a key-value store aimed at large schema-less documents.
Analytics and IoT
HDInsight is a managed Apache Hadoop (map/reduce), Spark, R, HBase and Storm service made "easy".
Azure Machine Learning is a set of machine-learning API's, allowing you to applying advanced analytics to a wide source of data (pictures, people, text etc.). As of Build 2016, this seems to be in the process of being rebadged "cognitive services".
Azure Stream Analytics gives you the ability to do real time processing of streaming data from huge numbers of sources.
Azure Data Factory is a set of data orchestration API's, allowing you to mangle data from different sources together (with tools for data lineage, ETL and so on).
Azure Event Hubs is a scalable pub-sub service for aggregating events from many sources.
Mobile Engagement is a set of API's for monitoring and understanding app usage on mobile devices.