What are the Android SDK build-tools, platform-tools and tools? And which version should be used?

I'll leave the discussion of the difference between Build Tools, Platform Tools, and Tools to others. From a practical standpoint, you only need to know the answer to your second question:

Which version should be used?

Answer: Use the most recent version.

For those using Android Studio with Gradle, the buildToolsVersion has to be set in the build.gradle (Module: app) file.

android {
    compileSdkVersion 25
    buildToolsVersion "25.0.2"
    ...
}

Where do I get the most recent version number of Build Tools?

Open the Android SDK Manager.

• In Android Studio go to Tools > Android > SDK Manager > Appearance & Behavior > System Settings > Android SDK


• Choose the SDK Tools tab.


• Select Android SDK Build Tools from the list


• Check Show Package Details.

The last item will show the most recent version.

Make sure it is installed and then write that number as the buildToolsVersion in build.gradle (Module: app).