The BaseGameActivity is the root of a game, that contains an Engine and manages to create a SurfaceView the contents of the Engine will be drawn into. There is always exactly one Engine for one BaseGameActivity. You can proceed from one BaseGameActivity to another using common Android mechanisms.
The Engine make the game proceed in small discrete steps of time. The Engine manages to synchronize a periodic drawing and updating of the Scene, which contains all the content that your game is currently handling actively. There usually is one Scene per Engine, except for the SplitScreenEngines.
An implementation of the IResolutionPolicy interface is part of the EngineOptions. It tells AndEngine how to deal with the different screen-sizes of different devices. I.e. RatioResolutionPolicy will maximize the SurfaceView to the limiting size of the screen, while keeping a specific ratio. That means objects won't be distorted while the SurfaceView has the maximum size possible.
A Camera defines the rectangle of the scene that is drawn on the screen, as not the whole scene is visible all the time. Usually there is one Camera per Scene, except for the SplitScreenEngines. There are subclasses that allow zooming and smooth position changes of the Camera.
The Scene class is the root container for all objects to be drawn on the screen. A Scene has a specific amount of Layers, which themselves can contain a (fixed or dynamic) amount of Entities. There are subclasses, like the CameraScene/HUD/MenuScene that are drawing themselves to the same position of the Scene no matter where the camera is positioned to.
An Entitiy is an object that can be drawn, like Sprites, Rectangles, Text or Lines. An Entity has a position/rotation/scale/color/etc...
A Texture is a 'image' in the memory of the graphics chip. On Android the width and height of a Texture has to be a power of 2. Therefore AndEngine assembles a Texture from a couple of ITextureSources, so the space can be used better.
An implmentation of the ITextureSource-interface like AssetTextureSource manages to load an image onto a specific position in the Texture.
A TextureRegion defines a rectangle on the Texture. A TextureRegion is used by Sprites to let the system know what part of the big Texture the Sprite is showing.
A PhysicsConnector manages to update the AndEngine-Shapes (like Rectangles/Sprites/etc...) when their physics representations "bodies" change. Once using Physics (and a PhysicsConnector) with an AndEngine-Shape you'd disable the Physics calculated by AndEngine itself, by calling setUpdatePhysics(false) to the Shape. Changes made to the AndEngine-Shape are not reflected in the Physics - you have to call the methods on the Body object you have used to create the PhysicsConnector.