We've got a lot of folks from the UK and other countries here on the board, and I know tipping is one of those bizarre, opaque American things that doesn't make a whole lot of sense. So here's a quick handbook on tipping situations for while you're on this side of the pond.
Restaurants
The most common situation where you encounter tipping in the U.S. is at restaurants.
- If you're using cash: Just leave the tip on the table when you leave. If you want, you can put the money in the folder that bill comes in (if it comes in one), or weight it down with one of the objects on the table. If you're worried for any reason that someone else will grab up the money, you can hand the tip directly to your waiter -- as an ex-waitress I can tell you I never minded this, unless my hands were full! Another thing people do sometimes is hand the waiter extra money when they pay the bill and tell them "Keep the change." This is also a good option if you're in a hurry, so that you don't have to sit there and wait for change.
- If you're using a credit card: After you give them your card to pay the bill and they run it, they'll give you a receipt that has an extra line above the signature for a tip. You can write in a tip on that line, add up the new total with tip at the bottom, and the restaurant then adjusts the final charge at the end of the night and pays the tip out to the waiter.* Make sure your math is right when you add it up, because by law, the restaurant can only enter the final total -- the tip line is just there to help you with your math. So if you add wrong, the waiter gets shorted. 
* I know that sounds like a scam, but it is a huge, huge, huge violation of federal law to mess with this. You do not have to worry about it. If a restaurant's trying to save money being shady, they're far more likely to understaff or water down the booze than to do something that might get the IRS's attention. If you're really worried, just save your receipt -- the restaurant will bring two copies, one labeled "Restaurant Copy" and one labeled "Customer Copy." If it's any indication of how rare it is for anyone to mess with tips, most Americans just ignore the "customer copy" and leave it behind.
How much should you tip at a restaurant?
Here's a basic guideline.
No tip = the nuclear option, usually only happens if you are completely and totally enraged at the service you received
<10% = usually a deliberate insult
10% = low, for when the service was just OK
15% = average; you're always safe with this number
20% = "good" tip left by happy and/or generous people
>20% = big tip
For large parties (usually 6 or more), most restaurants have a policy of adding a mandatory tip to the bill. If they do this, it's usually written at the bottom of the menu on the first or last page, and they will also give the percentage. Sometimes there's a sign just inside the restaurant with the same information. The tip will be on the bill already when they bring it to you. You never have to add more money on top of this tip unless you feel like doing so.
Bars
If you're sitting at a table and having drinks brought to you by waitstaff, you tip just like you would at a restaurant, when they bring the bill at the end.
If you're sitting at a bar being served by the bartender, $1 or $2 per drink is standard. If you want to make sure you have the bartender's attention the whole time you're there, give a $5 or $10 for your first drink, or at a posh bar or busy/popular club, drop a $20.
If you don't have any small bills, bartenders are usually pretty nice about breaking bigger ones unless you're at a snotty place.
- If you're paying cash, give the bartender his tip when he brings you your change for each drink, or you can do the "Keep the change" thing as described above. If it's crazy busy and he's already run away to serve another customer, you can just leave it on the bar in front of you.
- If you've given them a card to run a tab, you can pay at the end, or you can hand them cash as you go. It's a little nicer to hand them cash as you go, so that they're not wondering whether you're going to tip at all, but if you don't have any cash, obviously you do what you have to do.
Coffee Shops (and sometimes similar food establishments like ice cream shops, bakeries, etc)
These places often have tip jars out. It's a nice thing to do but it's not expected or required. Whatever amount you feel like giving is fine. People often dump the coin change from their purchase in the tip jar just to avoid having to carry the coins around. More than $1 is usually considered a big tip unless you had a large or complicated order.
Hotels
- Bellhops/Porters: These guys are kind of like waiters in that the majority of their income is tips. A LOT of people use them and don't tip them, but it's a jerk thing to do. (It's OK to not use them.) $1-5 per bag, minimum of $5 total, is a good guideline. Give it to them once they've brought all your bags to the room.
- Housekeeping: It's nice to leave $5 in a conspicuous spot when you leave the room for the day. Leave more if you've left a big mess or are obviously taking all the toiletries. If you can, also leave a note next to it saying that it's a tip for housekeeping, just to make it absolutely clear you're not one of those guests that randomly leaves cash lying around the room (you would be surprised what stupid things people do in hotels)
- Concierge: If they're just answering basic questions (which way to the X, what time does your restaurant close, etc), no need to tip. If they're doing research for you or making phone calls or setting up other services for you, tip. You can do this one of two ways -- $5-10 every time they help you out, or a lump sum of $20-50 around the time you check out.
- Valet Parking Attendants: On the off chance you've rented a car and you're having the valets take care of it, slip these guys $2-5 when they bring it up for you.
Food Delivery
If you're staying at a rental house or some other place where you can call a restaurant and have food delivered to you, tip the delivery guy $5 when he brings the food. More if it's a huge order.
That covers the tipping situations you're most likely to encounter. Any other questions or situations, post below and we'll try to help!