If you're visiting the US for more than a very short time, traveller's cheques are a convenient and safe way to carry money. Americans have been using them for a long time, even when travelling within the USA, and anyone with a bank account can usually deposit a traveller's cheque as though it were cash (provided that it bears their name as the payee, and it has two matching signatures).
But remember that in the US the traveller's cheque (or "travelers check") must be in dollars. No other currency will be accepted (except, perhaps, in a bank in some circumstances).
The assumption is that the original signature is guaranteed to be that of the legitimate owner of the cheque, and signing in front of the cashier with a matching signature proves that you're the same person.
It's as though you'd paid cash, and you'll get change if it's due.
If cheques get lost or stolen, and you report this as soon as you realise (along with the cheque numbers) they will be replaced - provided, of course, you've signed them once. If you've signed them twice, or not at all, then they probably won't be replaced. While they have the convenience of cash, they're fairly well protected if you lose them.
In theory, anyone with a bank account can deposit a traveller's cheque, but don't assume that everyone will be happy to accept one, particularly one of high value. It's wise to ask first, particularly before you sign it!
If you see a sign "No Checks", it usually refers to normal bank cheques and they may well still accept a traveller's cheque.
Some places may ask for ID before accepting the cheque. Sometimes this is because this is simply store policy (which may apply to credit card sales too), other times it's because the cashier hasn't encountered a traveller's cheque before and is treating it as a normal bank cheque. Your passport is a good form of ID, but the UK picture driving licence is a much better way (and avoids the need to carry your passport all the time).
Most traveller's cheques are issued by American Express and Thomas Cook, but Visa traveller's cheques are also available. Other financial institutions also issue them, but they're likely to be less readily accepted.
Always keep a note of the serial numbers of the cheques, and mark those that you've spent so that in the event of your losing them you can identify exactly which ones have gone. If you don't do this you may find it tough and slow going (and with some banks, impossible) to get your money back.
The change that you get after paying with a traveller's cheque can be a useful top-up for your cash. But try to avoid 't deliberately using high value cheques for small value purchases as a means of doing this, at least not without asking first - it's not generally liked by the stores. Instead, you can convert a traveller's cheque to cash at most banks and "check-cashing" facilities (although some may impose a charge, and it's wise to ask first).
If you choose to pay for an internet or mail-order purchase with a traveller's cheque through the mail, bear in mind that it must be properly double-signed and dated, and with the payee's name clearly written on it. Also, if the cheque gets lost it may be difficult to recover the cost from the cheque-issuing company, but it does at least make it easy for the person at the other end.
Finally, I should stress that the above is based on personal experience and a bit of research. It should be read as a general guide, and if you need to rely on any of the above, check the terms and conditions that come with the traveller's cheques, or with your bank.