I want to understand the difference between the following services when it comes to hosting api's on Azure
application gateway, api gateway, api management, api apps, app services
A simple and understandable example would be helpful as these services can get really confusing.
Azure Application Gateway and Azure API Management are no hosting solutions. They are solution to, simply put, manage traffic. Think about topics like Protection, load balancing, SSL termination, Header modification, Routing etc. They could very well be combined. See the docs for some guidance.
Azure Functions and Azure Web Apps are 2 hosting options, the former is a serverless offering that allows pay-per-use and Web Apps require an App Service Plan for which you pay a fixed price based on the provisioned resources.
api apps and api gateway you mention I cannot map to an azure service.
Here is a decision tree that can help clearify things regarding hosting options.