How does it work?
At its most basic re-steeping simply requires setting the strained leaves aside until you are ready to brew a second cup. All of the water should be removed from them in order to avoid bitterness. Every tea is a bit different but in a lot of cases you might increase the brewing time on the next round in order to extract enough flavor. If you're brewing in a western fashion (infuser basket or larger teapot), most unflavored teas can yield at least two to three infusions. Eastern style brewing (a la gongfu with gaiwans or yixing teapots) is a whole other story. I've had puerh and oolong that gave well over 10 infusions. The key difference is that a higher volume of leaves are used with a smaller volume of water, extending the life of the leaves.
How long can I let the leaves sit? Can they be brewed the next day?
It's really a matter of personal opinion but I try to use the leaves within a few hours. When I do leave them sitting, I'll usually put the cover on the gaiwan to make sure that they don't dry out. Once the leaves have dried out there really isn't a way to revive them.
There have been a few times where I had just started a session with a tea that I knew had a lot of infusions in it. Isn't it the worst when life interupts good tea? In that case I placed the closed gaiwan in the refrigerator, making sure that there was no sticky food inside that would ruin my tea. When I was able to return to the tea I made sure to do a few quick rinses to get the flavor back where it should be.
I would not be inclined to use leaves that have been sitting for longer 24 hours. Since they are moist used tea leaves make an excellent breeding ground for mold and bacteria. As heartbreaking as it is to throw out seemingly good tea leaves, I just don't think it is worth the risk. If I don't think that I'll be able to get back to drinking within that time frame, I'll use the leaves to make a batch of cold brewed iced tea instead. Just throw them into a mason jar full of cold water and place it in the fridge. I've found that oolongs work really well for this.
Will it work for flavored teas or tea bags?
Unfortunately flavored teas and tea bags are usually a one shot deal. Almost all of the flavor goes into the first cup so there will be very little left for a second one. The one exception to that might be full leaf pyramid style tea bags. As a general rule the smaller the leaf particles are, the less likely it is that you'll be able to re-steep it.