A scene (implemented with the Scene object) is a more or less independent piece of the app workflow. Some people may call them "screens" or "stages". Your app can have many scenes, but only one of them is active at a given time.
For example, you could have a game with the following scenes: Intro, Menu, Level 1, Cutscene 1, Level 2, Winning cutscene, losing cutscene, High scores screen.
You can define every one of these scenes more or less as separate apps; there is a bit of glue between them containing the logic for connecting scenes (the intro goes to the menu when interrupted or finishing, Level 1 can lead you to the cutscene 1 if finished or to the losing cutscene if you lose, etc.).
A scene is described in cocos2d as a tree of CocosNodes where the root is a Scene node, the most near descendants usually are Layers, and they hold and organize individual elements.
Example for the main menu screen:
main_menu_scene : Scene node holding all the main menu elements
animated_background : Layer depicting an animated background
static_background : Sprite with a nice draw covering all the screen
far_trees : Layer holding the most distant trees
tree_1..tree_k : Sprites showing trees
birds : Layer holding flying birds
bird_1..bird_n : Sprites showing birds
near_trees : Layer holding the most near trees
tree_1..tree_m : Sprites showing trees
main_menu : Menu, a Layer subclass provided by cocos that handles all the behavior
related to menu (key listening, highlight, select...)
item1 : MenuItem , 'play'
item2 : MenuItem , 'options'
item3 : MenuItem , 'quit'
There is also a family of Scene subclasses called transitions (implemented with the TrasitionScene object) which allow you to make transitions between two scenes (fade out/in, slide from a side, etc).
Since scenes are subclass of CocosNode, they can be transformed manually or by using actions. See Actions, Transformations and Effects for more detail about actions.