So here it is. Your starter has been bubbling away, becoming ever more alive over the last 10 days. You've nurtured it, fed it, talked to - oh. May be that's just me. Anyway, whatever, hopefully it's bubbling away, smelling beery and fermented, a little sour, but not unpleasant. What are you going to do with it?
Well, the first thing you need to do is refresh it. Now don't be alarmed, but after all that, you are going to get rid of about half of it - give it to a friend, perhaps: maybe someone who's been interested in what's been going on on your kitchen work top for the last few days? What ever, give it a mix, discard about half, add some more flour and water, give it another mix and watch it go. Your starter is now fully grown and you can start to bake with it. if you do give it away, the lucky recipients will need to give their newly acquired starter a feed of flour and water and they too will be ready to bake.
If you keep it out in the kitchen, you still need to feed it every day or so with some flour and water, and the idea is that you won't need to 'throw any away' (although you can of course give some to your friends who want to try it out too - just remember to top it up again) because you'll be using it fairly regularly to bake with. However, if you know you're not going to bake with it for a while, you have a few options.
Hugh F-W suggests mixing it down to a stiff paste with flour only, after which it will keep on a work top for 4 days before you need to re-feed it with flour and water. You can also now keep your starter dormant in the fridge for up to 2 weeks, in the same way - add flour, but not water, to make a stiff paste, cover it and put it in the fridge. Simple. In preparation for using it again, you need to get it out of the fridge and bring it up to room temperature. Give it a good stir and feed it (flour and water) and let it have a couple of hours to bubble up again.
I am lead to believe that you can keep it dormant in the freezer for even longer, but I have never tried it, and surfing around the Googlesphere, I find that you can also dry your starter (check out the comment by Midnight Baker).
Sometimes, you may find that it separates and looks a bit odd, but usually a good whisk to get some air into it will bring it bubbling back to life. It's quite hard to actually kill it off - you do need to keep an eye on it and feed it regularly if it's out on the work top, and if you are keeping it in the fridge, even if you don't bake with it, you probably need to get it out once every 10-14 days, bring it up to room temp, feed it and give it some bubble time.
Posts to follow shortly on actual baking - with apologies for the short delay if you hit the 10 day point over the weekend and were desperate to bake. I got distracted by marmalade...
And to recap, if you are coming new to the sourdough thing at this post, I've posted about getting a starter going, which you can read here. I've also written a post about what you might need (or not need) to bake your bread, which you can read here.