Dream thief Cobb (Leonardo DiCaprio) steals information and ideas from people's minds by infiltrating their dreams. However, he is tasked by Ken Watanabe's Saito to commit Inception — planting an idea into somebody's subconscious mind through a dream.
To do this, Cobb assembles a team of dream thiefs, starting with dream architect Ariadne (Ellen Page). A dream architect's job is to create the world of the dream itself — the subject (in this case, Fischer) is then brought into the dream and they fill the world with their subconscious.
So Ariadne creates the world of the dreams, while Cobb's team use dream-sharing technology so all of them can share the same dream at the same time. They do this by using a machine, as well as special medication.
Once they've achieved this, they hatch a dream-within-a-dream plan involving three layers of dreams.
Note: they use things called "totems" to determine if they are in a dream or in reality. Cobb's totem is a spinning top, which will keep spinning if he is in a dream but will stop spinning if he is in reality.
With the mission a success, Cobb is allowed back into the US, free of his criminal status. He visits his young children for the first time in years, and spins his spinning top totem to see if he is actually dreaming it.
But Cobb doesn't stay to see if the top falls or not, but instead joins his children. The final shot of the movie is of the spinning top spinning and possibly wobbling.
However, whenever Cobb is in a dream throughout the film he is seen wearing a wedding ring. When, in reality, he isn't. In this final scene, he is ringless, suggesting that the happy ending is real.