1 Corinthians 13 is probably the most well-known extended passage of the Bible. It has been read at countless weddings and other events, and for good reason: it is an amazing exposition of what love is. Amidst all the weddings and Christian love letters and everything else, we still need to remember that Paul wrote this for a reason, and it exists within a letter, not simply on its own:
"The key to understanding chapter 13, then, is to keep it in its context. Whatever inspiration it may have as a self-contained poem or hymn to love, Paul intended it to be used to help solve the specific problem of the destructive manner in which the Corinthians were using their spiritual gifts...
...If it is more excellent than even the greater gifts, then love itself cannot be a spiritual gift. Rather it represents the cardinal Christian virtue, the first on the list of the “fruit” of the Spirit (Gal. 5:22), which must be present with all the gifts if they are to be used in ways that will please God and have eternal value." 
This passage certainly has meaning and significance in the broader context of the Christian life, but its main purpose pertains to understanding and correctly practicing spiritual gifts. We need to remember that in order correctly follow the line of thinking in this letter. That being said, I'm going to spend the rest of the post talking about the meaning and implications of the "Love Chapter" in the broader sense of the whole Christian life.
One of the important things to remember when reading this passage is the main word: love. The problem is that while our culture has only one primary word with a number of different meanings, the NT has three main words it uses for "love": agape (Christian love for others), phileo (the type of love between a brother or friend), and eros (physical/romantic love). The word used in this passage is the first one. Why is this important? We must remember that this passage is speaking in terms of "agape" love, while our society defines love almost solely in the sense of "eros" love. I would argue that this passage isn't seeking to show us how to have a good relationship with a boyfriend/girlfriend. It's simply not even using that word. Our society has conditioned us to think of love mostly or only in that sense, whereas this passage is talking about a foundational kind of love for all that one builds the Christian life upon. What is THAT kind of love like?
"It has often been observed that one could substitute the word “Jesus” for “love” throughout verses 4–7. Indeed, as the only sinless person in human history, he provides the perfect model for helping us to understand what patience, kindness, lack of envy, and so on, are. In so doing, we also guard against misinterpreting these attributes. If Jesus was all-loving, but could clear the temple in righteous indignation (Mark 11:15–18) or unleash a torrential invective against the hypocrisy of the conservative religious leaders of his day (Matt. 23), then our concept of love must leave room for similar actions...
...Yet one of our greatest problems is defining love. Popular culture—in literature, music, advertising, and the visual arts—uses the word to mean just about everything except what the Bible means by it. So even Christians are easily misled into thinking love is primarily a feeling, something you fall in or out of. We equate it with lust or sexual intercourse itself, speaking of one’s “lover” (unless one is properly married, when the term actually would be appropriate) or of “making love.” But in this chapter, as throughout Scripture, love is first of all an action, an unconditional commitment, a promise that is never broken."  (emphasis mine)
That's a VERY important distinction to make: love is not primarily a feeling, but an action. This is the basis of the Biblical understanding of this love. You cannot building a life upon feelings, but you can build it upon action and commitment. Again, our culture teaches something different:
"Society confuses love and lust. Often, so do believers. Unlike lust, God’s kind of love is directed outward toward others, not inward toward one’s self. It is utterly unselfish. This kind of love goes against natural inclinations. It is possible to practice this love only if God helps us set aside our own desires and instincts so that we can give love while expecting nothing in return. Thus the more we become like Christ, the more love we will show to others." 
Lastly, some important thoughts about the closing remarks about faith, hope, and love:
"Faith is the foundation and content of God’s message; hope is the attitude and focus; love is the action. Faith informs action; hope influences action; love is action. When faith and hope are in line, you are free to love completely because you understand how God loves. Does your faith fully express itself in loving others?" 
Love doesn't just demand action, love IS action. If you're at all unclear about that, spend some time reading 1 John. Are you loving others, or simply trying to produce feelings?
 Ibid, 262, 264.
 Bruce B. Barton and Grant R. Osborne, 1 & 2 Corinthians, Life Application Bible Commentary (Wheaton, IL: Tyndale House, 1999), 188.
 Ibid, 194.