An event that is a beginning; a first part or stage of subsequent events.
But the movie talks about humans dreaming & lucid dreaming etc.
So, why is the movie titled Inception?
The technique is specifically named in the film, as the literal meaning of the word matches well with it. Following is a dialogue from early in the film, for reference:
COBB: What do you want from us?
SAITO: Is it possible?
ARTHUR: Of course not.
SAITO: If you can steal an idea from someone's mind, why can't you plant one there instead?
The remainder of the film deals entirely with a specific plot to do so, its planning and execution and ambiguous success. Saito intends to disrupt a power conglomerate from becoming a virtual monopoly, using inception on the heir apparent to the conglomerate owner's estate.
As you explained, Inception is an event that is a beginning; a first part or stage of subsequent events.
In this movie, lead protagonist Dominic Cobb(Leonardo DiCaprio) has got a task to implant an idea into Robert Fischer's subconscious mind by going into his dream. And that person will have a life and act according to what has been implanted into his mind.
Therefore, in this case, inception is just an activity to plant an idea into someone else's brain to start a chain of events.
They're planting an idea in someone's mind by convincing them that they came up with it themself -- the "inception" of a thought; where does it come from? What really puts it there?
That's the theme that all the main characters struggle-with, with their own life choices and the consequences of others' choices that they have to deal with, or the tasks they're trying to complete.
It revolves around the central question of "what started this?", because that could be how you answer "how did we get here?", which ultimately also leaves you wondering "what do we do next?".
Nij's answer covers well the in-universe explanation of what "inception" is in the context of the film: using the dream-tech to enter someone else's dreamworld and plant an idea that they come to believe is their own. This is explicitly exposited by Saito.
However there also exists a meta-level to the titling and, by extension, the theme of the whole film. Inception is very much like some of Nolan's previous films in that it tries to evoke aspects of the topic at hand; both Insomnia and Memento are prime examples of this, evoking feelings of the titular insomnia and the confusion of disordered memory suffered by their protagonists.
Inception goes one step further, concerning itself with the idea of lucid dreaming and the ability to (at least feel like you) control your dreams. Cobb's dialogue with both Ariadne- when he's training her in dream construction- and Fischer- when he's conning him into believing he's his dream security- are actual techniques for lucid dreaming: notice the strangeness of the dream, always be questioning reality, question how did I get here? Aside from necessary film techniques that could be interpreted as dream-like (cuts, for example) and the things explicitly shown to be part of the dream, the film's ambiguity as to the outer layers and whether Cobb's entire experience is a dream is left ambiguous.
This serves the main purpose of the film: to incept its audience with the ideas that promote lucid dreaming. The film experience lulls people into the experience of dreaming, but leaves them with the questions of "what is reality?" and "how can I be sure I am not dreaming?" While we don't have nearly as much uncertainty about our reality as when we're awake- mainly due to the equivalent tech not yet existing- the questioning should be sufficient to trigger lucid dreaming in some people.
Wiktionary defines inception as
The creation or beginning of something
Dom Cobb (Leonardo DiCaprio) is the only one that is convinced that in constrast to stealing information through dreams it is also possible to implant information.
He explains that this cannot be done directly though, but that one must plant a seed in the form of an idea or feeling that then will grow into thoughts and finally change that persons behavior.
Thus Inception refers to this initial implanting of an idea.
He knows this because...
..he tried that method with his wife and implanted her with the idea that the world she lives in is not real. He did this to get her out of the dream world. But that idea developed to such a strong conviction that even when she was in the real world she still was convinced that it was a dream and killed herself to "wake up".
Thank you for your interest in this question.
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