It's a good thing I read up on this passage because I learned a couple of things I didn't know before. First of all our English translation which reads "he will hate the one [master] and love the other [master]" comes off a little too strongly. It's not so much about hating and loving as we think of it, but more about having a preference for one over the other. "He will end up loving one master more than the other" is closer to what Jesus is saying. It's similar to the confusion over Jesus' saying, "If anyone comes to me and does not hate his father and mother..." The Lord is asking you to prefer him over your parents, not hate your parents in the way we normally think of that term.
Secondly, I learned that mammon can best be translated "material possessions," whereas I had always thought it was a synonym for "money." "Money" is apparently too narrow a meaning because "mammon" refers to your possessions in general. It hits on the problem of materialism, which is everyone's problem regardless of how rich or poor you are.
So now the passage makes a lot more sense. "No one can serve two masters" means that no one can be devoted to two masters. Sure, you can serve two masters in terms of fulfilling your duties to both of them. We do it all the time. For example, at work you can have two bosses, one in lower management and one in upper. But you will always compare the two and have a preference for one over the other. "For either he will hate [love to a lesser degree] the one and love [to a greater degree] the other, or he will hold to one and despise the other."
Therefore, Jesus says, "You cannot serve God and mammon." Jesus knows what a stronghold material possessions have upon us. We have to surround ourselves with lots of things to feel secure. We derive feelings of worth from our possessions. We put our trust in the number that shows up on our bank account statement. Security, worthiness, trust--see how easily all those feelings that should center upon God get sucked into the vortex of material things? That's what Jesus is saying. You cannot serve both God and materialism. You tend to attach yourself to either one or the other. You tend to invest all your hope and self-worth and general feelings of well-being into either the God who loves you and gives you all things generously, or into the things themselves--and then you sit surrounded in the middle of your pile as if it were a great fortress against the uncertainty and tragedy of the world.
Jesus knows that we need material things like food and clothes and shelter but it's just that we need to be aware of the two masters principle. Our hearts tend to cling either to the Giver or to the Things Given, and we have to be on guard about that. We have to focus on the Giver always, which Jesus is going to talk more about in our next passage.