Thomas A. "Tom" Anderson (alias Neo):
The word 'Neo' is an anagram not only of the word 'one,' but also for 'eon'. Neo is also Greek for 'new,' suggesting messianic overtones for his mission in the Matrix. In Gnosticism eon's, or aeon's, is a kind of superhuman being with supernatural powers.
Neo has carried since his inception the Matrix's source code, known as the Prime Program. This gives him the ability to freely manipulate the simulated reality of the Matrix, similar to the authority a system administrator has over a given system. However, he mainly manifests these abilities as various superhuman powers.
The power Neo exhibits most often is telekinesis - he seems capable of manipulating any object in the Matrix through will alone. By focusing this ability upon himself he can fly at amazing speeds and jump great distances. Whilst his speed is never specified, he flies from The Merovingian's mountain manor to the highway "500 miles due south" in fewer than 15 minutes (2,000 miles per hour), implying that his "cruising speed" is in excess of three times the speed of sound. His speed of flight is further exemplified by his ability to "outfly" explosions and the fact that as he flies, objects around him tend to twist, catch fire, or smash; the sonic boom left in his wake has the power to overturn rows of heavy vehicles.
Further, Neo possesses superhuman strength and agility, and is able to brush off attacks that would kill or disable a normal human being with relative ease. His reflexes are great enough to dodge bullets in a manner similar to the Agents (though he usually stops them outright with his telekinesis). However, his strength and endurance have limits. Even though he could block a sword with the edge of his hand, he received a small cut, indicating (as the Merovingian pointed out) that he was still human despite his powers. His endurance is also finite: when confronted by masses of Smith clones in the second film, Neo was forced to escape rather than continue fighting, and upon exiting the Matrix, he appeared visibly tired and out of breath.
Neo has displayed the ability to heal people within the Matrix, even to the point of resurrecting the recently deceased (though the extent and limits of this ability are not known).
A more subtle application of Neo's ability is a heightened awareness of the simulated nature of the Matrix, enabling him to easily detect Agents, Exiles, and other anomalous phenomena (such as the presence of explosives). He is even capable of a limited form of precognition. The ability to see the Matrix's inner workings has been referred to as "Code Vision." This has been described as "seeing the world without time", or visualizing possible futures like a chess player looking several moves ahead.
In addition, Neo (like most Red Pills) utilizes the Resistance's combat programs, which grant combat abilities equivalent to a martial artist with decades of experience. However, Neo is able to run an unprecedented number of these programs simultaneously; he is therefore a master of every known martial art and fighting style, including Kung Fu, Jujitsu, Karate, Eskrima, Drunken Boxing, and Taekwondo (the Nebuchadnezzar ship operator Tank uploads each of these martial arts programs (and more) to Neo's brain during his training in the first movie) (However, in training for the series, Keanu Reeve's fighting arsenal, as well as the entire cast's, emphasizes Wushu.)
In the real world, Neo has some degree of wireless connectivity with the Matrix system: he eventually learns to sense the presence of the Machines, and even interfere with their function. This capability actually overwhelms him at first, causing him to involuntarily jack into Mobil Avenue Station without the equipment normally used by the Resistance. According to the Oracle, "The power of the One extends beyond this world [the Matrix]. It reaches from here all the way back to where it came from." Apparently because of his status as The One; he has a direct connection to the Source, and can therefore affect everything connected to it. Additionally, after he is blinded in the fight against Smith/Bane, Neo begins to perceive everything connected to the Source, including the Machine City itself, as silhouettes of golden light.