A: 101 actually appears a number of times in all three films, and though no one knows exactly what it means (if anything), the popular explanation is this: 101 is binary for 5 (which to a computer is the sixth number because computers count from 0), and the events of the three films take place in the 6th interation of the Matrix. Neo is the sixth One, this is the sixth time the machines have destroyed Zion, etc. We learn that this is the sixth iteration from The Architect's conversation.
Another popular explanation for its prominence: 101 is a direct reference to George Orwell's 1984. Room 101 is the place in which a persons greatest fear of all is enclosed. Thanks to Daniel Bryan-Curnow for the Orwell reference!

A: In the first scene Trinity and Cypher are on the phone. Cypher (who is later revealed as a traitor) has given the Agents enough information for them to trace the call and determine Trinity's location. Additionally, it's interesting to note that successfully locating Trinity was the evidence the Agents needed to believe the informant was real and could therefore be useful to them.

A: The book is Simulacra and Simulation, by Jean Baudrillard, and it's one of three given to Keanu Reeves before reading the script. For more on this, see Matrix Inspirations.

A: Morpheus is watching the green raining code screens on the Nebuchadnezzar. The code is the visual representation of the Matrix. He's able to decode what he sees on the screen, giving him a full view of Neo's office building, locations of Agents, the existence of the window washer platform, etc. Think of it like unlimited security camera footage.

A: He believes Neo is the One that the Oracle prophesied. Find out more about the One.

A: The red pill changes the way Neo's body is interfacing with the machines. This change allows Morpheus' crew to pinpoint his physical location in the powerplant in order to free him.

A: Trinity's talking about Neo's feeling that the world is unreal, that there's something missing. He's been searching for something, and if he doesn't listen to her now, he's going to return to his old life where 'he's not sure if he's awake or still dreaming'. For a complimentary perspective on this question, see Contributions: Matrix Reader Theories.

A: It's never explicitly stated, but it appears in waking Neo up, they are able to convince the machine's system that he is garbage. After he breaks out of the cell he is in, a component of the system removes the cable from the back of his head, and his cell is flushed into the waste disposal system. This leads to the sewers, which is where the Nebuchadnezzar is waiting. They pluck Neo from the waste and hoist him to their ship.

A: Zion is where free humans live. Far below the destroyed surface of the earch, the citizens of Zion are both former inhabitants of the Matrix who've been freed, and also 100% home-grown humans that have never been slaves to the Matrix.

A: The Nebuchadnezzar is a hovercraft flying through the abandoned subway and sewer lines beneath the destroyed surface of the earth. Somewhere in the sewers are tunnels leading to Zion, very far below the surface.

A: After man created Artificial Intelligence, the intelligent machines wanted their own society. Misunderstandings led to disagreements, which led to a common human response to things we don't understand: we tried to destroy them. They fought back, and in effect, they won. The fight included nuclear weapons which pretty much destroyed the surface of the earth. Find out more about The Second Renaissance.

A: If Morpheus is right, and Neo is the One, he should be able to bend the unreality of the Matrix to his will. He should be able to do things the rest of them have only dreamt of. He should be faster, stronger, almost a superhero in the Matrix. This is where they hope to see the first glimpse of his capabilities.

A: Cypher is setting up his meeting with Agent Smith in the restaurant. He's now committed to exchanging Morpheus for his own reinsertion back into the Matrix.

A: No one knows for sure how he did it, but a few scenes previous when Neo comes in and startles Cypher (when they share the drink), he was setting up the meeting with Smith in the Matrix. It's likely he also set some sort of timer/program to make the call to jack out once the meeting was done.

A: The spoon exists only in the Matrix, which really means it doesn't exist. It's a lesson for Neo, to help him realize that manipulating the Matrix isn't about focusing on an object and trying to change it. The object doesn't exist, so he can't change it, he has to change himself. Metaphorically, it's all in his head.

A: As Morpheus later says: she tells Neo what he needs to hear. What path would he have followed if she'd said he was the One? Would he have been cockier, more arrogant, less careful? Would things have ended the same way? She apparently knows the future, so she knows what she needs to tell him to get him where he needs to go. Find out more about Neo's conversation with the Oracle.

A: A manufactured accident to allow Cypher to get back onto the Nebuchadnezzar alone. There he can kill the rest of the crew by unplugging them, leaving only Morpheus alive and reducing the chance of interference from them.

A: They don't really want Morpheus, they really want the access codes to Zion's mainframes. Breaking into the mainframe will give them enough information to find and destroy Zion, thereby destroying the human resistance. Morpheus knows these codes and they intend to get them out of his head by any means necessary.

A: He doesn't want the other Agents to know what he's saying to Morpheus. The Agents seem to operate collectively, finishing each other's sentences at times, and certainly knowing what each is thinking. Smith's actions may largely be symbolic to show that he's able to turn this collective consciousness off, and in this case, has a reason to. His discussion with Morpheus implies that he is much more "human" than his colleagues - he shows frustration, exhaustion, impatience, and hatred of the Matrix itself and his place in it.

A: Trinity's talking about the fact that the Oracle told her she would fall in love with The One. Everything else the Oracle told her has happened, and it seems like this last piece of the puzzle is falling into place because she has to accept and admit that she loves Neo. This ties directly into what she tells Neo's lifeless, jacked-in body before she kisses him.

A: The Agents probably don't know that the machines know where Zion is. Agents are simply programming constructs with a set purpose. They have little or no knowledge of the overriding system or their place in it. They just do what they're programmed to do - and in this case, they're programmed to seek out access codes. It's all part of the control of the machines.

A: Not exactly, though interpretations differ slightly. Simplest explanations are always best: Neo is the One. Trinity was told by the Oracle she would fall in love with the One. The One will save humanity and help overthrow the machines. Once Trinity voices her love to Neo, it's like the last link in the prophecy chain - he must be the One, so he can't die until he's done what the One is there to do. Once that link clicks into place, it's apparent that Neo can do whatever he wants in the Matrix.

A: Now that he truly believes he is the One, Neo is beginning to use some of his power over the Matrix. He can now see the Matrix for what it is, and this will lead to additional abilities to manipulate it as he sees fit.

A: Neo is talking (possibly metaphorically) to the machine mainframe. He's talking to the Source, to the power behind the machines, to whatever sends the Agents after him. And he's telling it/them that he's done running, that he believes the Matrix can be a prison for humanity, or possibly something more, and finally that he wants peace.