What Is The Matrix

Morpheus: Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.

Neo: We are now inside a computer program!
Neo: This..... This isnt real!
Morpheus: What is 'real'? How do you define 'real'? If you're talking about what you can feel, smell, taste, and see then 'real' is simply electrical signal interpreted by your brain!

Morpheus: What is the Matrix? Control.
Morpheus: The Matrix is a computer-generated dreamworld built to keep us under control to change a human being into this\(a battery)...
Neo: No! I dont believe it! Its impossible!

(A boy bends a spoon without even touching it)
(The boy hands the spoon to Neo)
Boy: Dont try to bend the spoon, thats impossible! Instead only try to realize the truth!
Neo: What truth?!
Boy: There is no spoon!

Trinity: The Matrix isnt real!
Cypher: I disagree Trinity: I think the Matrix could be more real than this world... (source)

The Matrix presents a version of an old philosophical fable: the brains in a vat (with some spice to make a cool film). Since its not easy to explain all aspects here, read more in the given link!