Full spoilers follow.
The job. Saito hires Cobb not to steal but to plant: Robert Fischer, heir to a rival energy conglomerate, must be induced to break up his father's empire — and must believe the decision is his own. Payment is not cash but Saito's ability to make Cobb's homicide charge disappear so he can re-enter the United States and see his children. The crew go three levels deep (rain-soaked city and van, hotel, snow fortress), with time dilating by orders of magnitude at each descent, and a fourth region below: limbo.
The buried truth. Cobb's wife Mal is dead, and he killed her — not directly, but by performing inception on her. Years earlier the two spent decades together in limbo, building a world; when Mal became unwilling to leave, Cobb planted a single idea in her mind, that her world was not real, in order to persuade her to die alongside him and wake up. The idea worked, and then it would not stop working. Back in reality she remained convinced she was still dreaming, and threw herself from a hotel ledge — arranging the scene, and her own legal declarations of sanity, so that Cobb would be charged with her murder if he did not jump with her. This is why he is a fugitive, and why the top was hers before it was his: the totem he uses to verify reality is an inherited object from the person his certainty destroyed.
Why the mission keeps failing. The crew's obstacles are only partly Fischer's militarised subconscious. The real saboteur is Cobb: his projection of Mal repeatedly invades the shared dreams, shooting Fischer at the critical moment on the third level and forcing the team to improvise. Ariadne, having secretly explored Cobb's memory elevator — a vertical index of remembered rooms, ending in the hotel where Mal died — is the only one who understands the risk, and insists on descending with him.
The resolution in limbo. Fischer is killed on level three and drops to limbo; Ariadne and Cobb follow to retrieve him. There Cobb finally confronts Mal and articulates the thing he has been refusing to say: the version of her in his mind is a shadow, an insufficient projection assembled from his memory, and he cannot keep her alive by continuing to build for her. The confession is the film's actual climax — the inception being performed on the audience's protagonist rather than on the client's target.
Fischer's catharsis. Simultaneously, the crew's design pays off. Fischer opens a safe in the snow fortress and finds his dying father's pinwheel and the words he needed: that his father was disappointed because his son tried to be him rather than himself. It is entirely manufactured, and it is the most emotionally genuine reconciliation in the film — the mechanism the movie uses to prove its thesis that a planted idea is indistinguishable, from the inside, from self-knowledge.
The kicks. The layers unwind in synchrony: the van hitting the water, the elevator drop Arthur rigs after gravity fails, the fortress detonating. Cobb stays behind in limbo to find Saito, who has died and aged decades there; Cobb reminds him of the deal, and the two return.
The ending. Cobb wakes on the plane. Saito makes a phone call. Immigration waves him through. His father-in-law meets him at LAX. At home he spins the top on the table — and then his children turn around and he goes to them, leaving the top spinning. The camera holds on it; it wobbles; cut to black.
What only becomes clear afterwards. The film has been an inception performed on Cobb himself all along, with Ariadne as the architect of his catharsis. The top was never his totem — it was Mal's, which means it could never have told him anything reliable. And the decisive detail is behavioural rather than physical: for the entire film Cobb watches the top compulsively, and in the final scene he walks away before it resolves. Nolan's reported position is that this is the answer — Cobb has stopped caring whether it falls, because he has accepted his world. The unresolved wobble is left for the audience, who are placed in exactly the position of doubt Cobb has just abandoned.