The primary conflict is between the machines and humans...
The machines (computers) control the world and use humans as energy sources to power their functions. The humans are kept sedated and fed a false, computer-created world (the matrix) that seems real to the sedated humans. However, there is another group of humans, outside the computer's power grid, that considers the sedate life captive humanity is kept in as nothing more than slavery. And, they are fighting to liberate the humans used to power the computers, the "slaves" to the computer. Zion is purposefully allowed by the machines to exist, since there are issues in the matrix -- required issues since humans, according to the trilogy's mythology, can not exist in a utopian, perfect world -- that will eventually destroy the matrix if the program is not rebooted by outside sources ever so often. There have been, the Administrator tells Neo, five previous "Ones", five previous times that Zion has been destroyed and rebuilt, five previous times that the matrix has been rebooted.
On a philosophical level, the conflict is between a difficult, but free life, and a fairly easy, but slave, life. Now we understand that a difficult free life is better than a difficult slave life, for at least freedom gives us the hope -- some would say illusion -- that conditions can improve through the choices that we make. But what if all of the slave's needs were at least adequately taken care of? What if the conditions are a comfortable life, but still slavery? As Cipher says to the computer-created agents at a restaurant while he agrees to turn over Morpheus, "You know, I know this steak doesn't exist. I know that when I put it in my mouth, the Matrix is telling my brain that it is juicy and delicious. After nine years, you know what I realize? ... Ignorance is bliss." And, the question that Cipher asks is, "Which is better? Gruel eaten in freedom or steak eaten in slavery?" Morpheus and Trinity believe that gruel eaten in freedom is better than the steak eaten in slavery. Cipher, who I personally believe would be the majority view, believes that steak eaten in slavery is better than gruel eaten in freedom.
A third conflict in the movie and one explored in the sites below, is the Capt. Kirk/Spock conflict explored in the original Star Trek series. It is the conflict between reason, hard science, mathmatic precision, rationality and logic (all of which Spock represented in the original Star Trek series) versus emotions and love (which Kirk represented). In the Matrix trilogy, Neo and Agent Smith are only pawns through which the conflict plays out. The conflict is between Oracle, who wants to see the war ended, and the Architect, both of whom help control the matrix. The Oracle, who has seen all five past matrixes destroyed and a new one created, is affected by human emotion and love. The Architect, even through the very end of the movie, is still coldly rational, but the Oracle appears to have prevailed in this final conflict.