First, a caveat: the Coase Theorem is an interesting thought experiment and a stepping-stone to further understanding and not intended as a real description of the actual world. Says who? Says Coase. Please keep that in mind.
Now: imagine that you've decided to go into the lettuce-farming business, and your kid sister has decided to make her fortune farming turtles. Your parents give you neighbouring patches in the back yard and you start sowing lettuce seeds in your little plot. A few months later your lettuce is coming through nicely... when you hit a problem. Late at night, those sneaky turtles are crawling onto your lettuce patch and pecking at your lettuce.
You deal with this the usual, grown-up way and go running to your ma. You explain that either your sister needs to build a fence around her turtles or she needs to pay you for the damage that will happen to your lettuce (you can't build a fence yourself, because you're incompetent, but your sister is a great builder).
There are two options for how your ma can respond: she can either decide that your sister is liable for any damages caused by her turtles, or she can shrug her shoulders and say that if your sister's turtles want to eat your lettuce then you'll jolly well have to let them.
And here's where we get hit by the Coase Theorem: Coase says that -- in a world with no transaction costs (we'll come back to this assumption soon) -- which set of rules your mother lays down will have no impact on whether or not your sister puts up a fence. It might impact the distribution of wealth between you and your sister, but it won't affect whether the fence goes up. At first this seems a bit weird -- surely if your mother makes your stupid sister pay for the stupid damages caused by her stupid turtles then she'll put the fence up, and otherwise not? That's why Coase is so interesting, because it rubs up strangely against our intuitions.
Suppose that raising the fence costs your sister $2. There are two cases we need to consider.
First, suppose that the damage to your lettuce over the course of this year's lettuce season will be worth more than $2 (to you) -- suppose it's worth $5. If your sister is on the hook for the damages then she'll spend $2 to put up the fence and avoid paying $5 in lettuce-damages. But what if she's not on the hook for the damages? Well: since you're about to suffer $5 of damages, it'll be worth your while to pay your sister $2-5 to put up the fence and avoid the damage to the lettuce (and worth her while to accept, because she'll make a profit). Either way, the fence goes up.
Second, suppose that the damage to your lettuce will be worth less than $2 (to you) -- suppose it's worth only $1. In the case where your sister is on the hook for any lettuce-damages, rather than putting up a $2 fence, she will simply pay you $1-2 for the damages -- and it will be worth your while to accept. What about the case where the damages are on you? Well, since you're only going to suffer $1 of damage -- and you'd have to pay your sister at least $2 to build a fence -- you'll just eat the losses (with some tasty home-made salad-dressing). Either way, no fence.
Now: if this doesn't sound too much like the real world, that's because of transaction costs. Imagine that, in order to get your sister to build a fence, you have to sign a contract and hire a lawyer (Billy from down the street) and he charges you a $6 fee to write the contract. Now, everything changes. If your damages are $5 and your sister is on the hook for them then she will simply build the fence, but if you are on the hook for the damages then you'll have to eat the cost because instead of paying your sister $2 to build a fence you have to pay $2 for her and $6 for Billy and $2 + $6 = more than your $5 damages. In the real world transaction costs are everywhere, and that means that the "raw" Coase Theorem rarely applies. However, the theorem still gives us a new jumping-off point to understand how the "rules of the game" will affect actual outcomes.
Wikipedia on the Coase Theorem, as always.
There's lots of disputes about if/when zero transaction costs ever apply in practice, but H. Elizabeth Peter's paper on Coasian divorce got a lot of traction. She presents empirical evidence that different types of divorce law in different U.S. states affected how assets were split in a divorce, but not how often divorce occured. This is in line with the predictions of the Coase Theorem in a zero transaction-cost world.