What's the deal with all these Microsoft "apps"

Here's my attempt.

PowerApps

Geared toward business analysts and specialists to create apps for mobile devices (excluding Windows phones) using an Office-like experience. No writing code.

PowerApps has two parts: client (also known internally as Studio) and the service (also known internally as the 'portal'). You install the client on your computer, and use it to create apps. The service (web.powerapps.com) is where you can also create apps, and you create any connections. These "connections" / "data sources" connect to different services, like SalesForce, Wunderlist, GitHub, OneDrive, Office 365, yada yada yada. See this list.

The client tool is similar to PowerPoint (imo), and has a function bar for expressions similar to Excel. When you go to add a connection, web.powerapps.com automatically opens, and shows you the list of all the connections.

PowerApps has no direct relationship to Azure or the Azure portal. If you're good with Excel or app creation in general (notice I didn't say app development), you need to checkout PowerApps.

App Service

A suite that includes platform services geared toward developers to create web apps, mobile apps, and API apps using different languages, like .Net, NodeJS, and more. App Service is an Azure offering, and therefore uses an Azure subscription and the Azure portal.

If you previously used Azure websites and mobile services, this is where you belong.

Logic Apps

Part of the App Service offering kinda. Geared toward developers and IT pro users.

It uses the Azure portal to create logic apps using a designer. You create a flow-like experience starting at the top, add different connectors for your "message", and then end your flow at the bottom. Logic apps uses triggers to get a message (to give you an example), and actions to do something with the message (e.g. save to OneDrive).

If you're a BizTalk user and want to connect your systems or services, this is where you belong. If you want to start creating business-like flows and don't consider yourself a BizTalk person, I encourage you to try logic apps. The user experience is much improved and easier to use compared to previous versions.

API apps

Geared toward developers that need to create their own APIs using ASP.Net, C#, Java, Python, and more. Your custom API apps can be consumed by Logic Apps. Go here for more specifics.

Connectors / Connections / API apps

Where to begin...  They were originally called "API apps; not be confused with API Apps in App Service. Well, we're now calling these "connectors", even the ones that don't actually "connect", like the BizTalk* connectors (Transform, XML Validator, ...).

These connectors are based off swagger files. PowerApps, Logic apps, Flow, and soon to be others all use the same swagger file. For example, if you're using the Salesforce connector in Logic apps, and the Salesforce connection in PowerApps, then you are using the exact same Salesforce swagger file. The difference is how the client (Logic apps vs PowerApps vs Flow) exposes the information in the swagger file. In logic apps, we use Triggers and Actions. In PowerApps, we use Functions. They are the same - just different naming.

Connectors use OAUTH, so you'll need an account and a password to connect.

BizTalk Services (MABS)

It was called AppFabric Integration, then it was Service Bus EAI & EDI Labs, then it was called Windows Azure BizTalk Services, and now it's called Microsoft Azure BizTalk Services. If you are considering an Azure solution for your BizTalk applications, then look at Logic Apps.