The first word in John 3:16 is "For". This is a conjunction, meaning it joins two things, the way "and" does. So you might guess I'm about to say something about the context, and you'd be right.
In this passage, Jesus is talking with Nicodemus, a Pharisee and member of the ruling council (John 3:1) who came to Jesus at night (John 3:2). Jesus is explaining how someone can enter the kingdom of God (John 3:5). The dialogue continues in the style of ships passing in the night, then Jesus says:
14Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up, 15that everyone who believes may have eternal life in him.Clear as mud, right? It looks like Jesus has gone from "kingdom of God" to "eternal life"; he seems to be using them interchangeably. And what's this about Moses and the snake? That's not a real popular Sunday School story, is it?
16For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. 17For God did not send his Son into the world to condemn the world, but to save the world through him.John 3:14-17 (NIV)
The snake story is from when the Israelites were in the wilderness, and they complained about Moses and about God. This wasn't a good idea:
Then the Lord sent venomous snakes among them; they bit the people and many Israelites died. The people came to Moses and said, “We sinned when we spoke against the Lord and against you. Pray that the Lord will take the snakes away from us.” So Moses prayed for the people.A lot has been written about that bronze snake (like doesn't it violate the Ten Commandments—the one about making graven images in the likeness of things on earth?) but what I think relevant here is how bizarre the whole thing is. "Look at the snake on the pole," a snake-bite victim hears, "and you'll be cured of the snake-bite." At that point a bitten man or woman (or child I suppose) has a choice: Believe the word of the Lord and live; or disbelieve and die. Why would looking at the bronze snake on the pole ("It looks like the thing that bit me!") cure the snake-bite? That's why I said it's bizarre.
The Lord said to Moses, “Make a snake and put it up on a pole; anyone who is bitten can look at it and live.” So Moses made a bronze snake and put it up on a pole. Then when anyone was bitten by a snake and looked at the bronze snake, they lived.Numbers 21:6-9
Fast-forward now about a thousand years: Jesus is saying that we're like the Israelite snake-bite victims in that we've been bitten and we're currently headed in a direction that leads to perishing. The good news, though, is that God has provided a cure: to believe in the exalted (lifted up) Jesus. Why would believing in this crucified man (the punishment looks like what we deserve for our sins) cure us from the deadly disease of sin? That's why I say this is bizarre too.
Here's my take on the answer: Looking at the snake, as opposed to rooting around looking for an anti-venin kit, would be an act of faith. By looking at the bronze snake—looking up at it high on the pole—the victim confesses their desperate situation ("I'm going to die from this snake bite") and their need for something miraculous. Refusing to look at the snake reflects either denial of the seriousness of their situation ("I'll be all right...") or disbelief of God's promise ("I don't have time for an art appreciation class right now").
In the above passage, the Bible tells us that we're in deep trouble; we're perishing. Really! We've got the inevitability of death and the meaninglessness of life, we've got dangers all around us, we've got isolation and alienation, and the worst part is that we can't even control our own selves (try going 24 hours, or even one waking hour, without complaining). To acknowledge the seriousness of the situation, and to believe what God says, means believing in Jesus. Refusing to believe in Jesus means either I think I'll be all right, or I think I have some other means to make myself all right.
Looking at the snake, or believing in Jesus, is an admission of my helplessness and an act of faith in God. This pleases God, as Hebrews 11:6 suggests: when we believe God exists and that he rewards those who seek him, we do that (i.e., seek him) and this pleases him. And that's what it's all about.
Now for a plot twist: In the desert, Moses made a snake from bronze, which took some work but I wouldn't call it an act of amazing love or sacrifice; at Calvary, God gave his one and only son, the sinless Jesus, as a sacrifice for us, so that we might have eternal life. This took a lot of love—amazing love, in the words of the song—he loved the world to such an extent that he sacrificed his only son.
And when this son, a free gift from God, was rejected and crucified—that is, when we humans were unloveable—God showed his amazing love, not only by sending his son, but by giving him up for us. So that whoever admits their desperate situation and acts in faith toward God by believing in Jesus, need not perish; we can instead have eternal life.
And that's what John 3:16 means.