There are two major areas of PaaS that the major CSPs have that you can use to replace a single tier of your web application: App Tier and Data Tier services.
EXAMPLE: Take our web application and focus on the Web and Business Logic tiers and have them deploy them through AWS Elastic Beanstalk or into Azure App Services. In AWS, your tiers are still deployed into IaaS based services that you are able to see and modify, but AWS is now managing things like OS patching and updates to your application level software as well as making sure that all the pieces are talking to one another and have the right configurations for Elasticity and Health Monitoring.
In the case of Azure, you would define ahead of time what level of service you want to use and then deploy the code. Azure abstracts away all the underlying architecture and manages the OS patching, updates to your application level software and makes sure that all the pieces are talking to one another. However, it is seen as a complete black box deployment with you not able to see the underlying VMs, Load Balancers, etc. Afterward, you would configure the service for things like Elasticity and Health Monitoring.
Data Tier – Within this area the CSPs provide a number of different options , called DBaaS, that you can leverage to replace your data tier while still providing either Relational or NoSQL based data storage and allowing you to focus on only what your application needs. As you would expect, there are many different options for relational databases that are supported within the different CSPs. Some of the options that I have seen are: SQL Server, Oracle, MySQL, PostGreSQL among others. Some of the CSPs have even created their own versions of one or more of the popular Open Source options. Each of the CSPs also provide a number of NoSQL options, such as Document based databases that are MongoDB supported as well as Graph, Tabular and In-Memory Key/Value pair based databases like Redis Cache.
In each case, the service is a fully managed deployment of the database requiring you to only worry about the database and the data within. All of the infrastructure and corresponding OS and software is managed by the CSP.