"No one can serve two masters; for either he will hate the one and love the other, or he will be devoted to one and despise the other. You cannot serve God and wealth. (Matthew 6:24 NASB)
The idea of 'serving a master' is to be enslaved to someone or something. It seems logical that if you were enslaved to two masters you would have a very miserable existence, even if one of the masters was good. In fact if the two masters have opposite intent the slave then has an exceedingly miserable experience. This is part of why divorce negatively affects kids so much, they begin life with parents who are 'one' who share the same purposes, discipline, commands, and expectations, and essentially the child has one master, the parental unit. Now when the split happens it all changes, each parents individual preferences in discipline come out, and their expectations or lack thereof come out, and now the child has two individual masters and it is impossible to live up to two separate standards. The parents who at one point had to reconcile all their ideas and methods concerning raising their child in order to function as family, once they sever the bond of marriage no longer reconcile these ideas and methods and the child now has two masters. (I am no psychologist, but that has to mess a kid up.)
Now as a believer you may potentially be stuck in the same boat. Wealth and God, they are two very different masters, that make two very different sets of demands on you. Can you both store wealth and give liberally? Can you give your time and energy to the selfish pursuit of riches and still give that same time and energy to the service of the Lord? You cannot serve two masters. Slavery to both God and money is an utter impossibility. Now many people who have acquired much wealth and make a profession of faith in Christianity are far more enslaved to wealth than they care to admit. They call the wealth a blessing from God, which it might be, but they never examine themselves to see whether or not that wealth is providing any spiritual blessing either to themselves or to others. How can you know if you are mastered by wealth and possessions? Where is your mind? Do you give more worry to your financial situation than to your godliness, or glory of God in the world? Do your possessions receive more of your time than the Lord? When you have a free moment to sit and do nothing, where does your mind typically wander, to possessions, politics, work, money, etc... or to the Lord? It is not difficult to know who your master is, just look at your life, look at your free time, look at your conversations, and know that you cannot serve two masters!
Notice that we are not called to despise wealth, just not serve it, if we try to serve God and wealth we will either love God and despise wealth, or we will love wealth and despise God. In either case our existence is miserable, if we love God and despise wealth, then finances become a great source of woe to us. Bills, and savings accounts, the next purchase, and so one become items of contention in our soul and generate misery in our lives. Our love for God causes us to loathe our other master, wealth. Of course if we love wealth, we view God as this miserable kill joy trying to take away all of our possessions, as though God is a constant restriction upon our lives. The ideal situation is that God be the sovereign master over all our affairs including 'wealth' at which point wealth is no longer our concern. Wealth in our lives becomes subject to God, it is no longer a burden to us, because we trust that our Master will provide for our needs, and he always does. Bills, tithing, expenses, etc... become an opportunity to celebrate the faithfulness of God as He provides the means to meet them. Finances which were a hard master to us, now become a source of great blessing to us. You cannot be a servant to both God and Wealth... you just cannot, that is unless you are OK with hating one and loving the other.