The Republic of Ireland ( Ireland Tourism) uses the Euro (€). The notes come in €500, €200, €100, €50, €20, €10 and €5 denominations.
NB: €100 notes and above may not be accepted or be carefully scrutinized in shops. 100 cents make up €1 and the coins are as follows 1c, 2c, 5c, 10c, 20c, 50c, €1 and €2. 
While foreign currency is generally not accepted, some large shops in places popular with tourists, will accept major foreign currencies such as dollars and sterling, but the rate you will be offered will generally be inferior to that offered by a bank or credit card issuer. 
In Northern Ireland, the Pound Sterling is used.  The currency is the same as that used in Great Britain but the 4 main retail banks each issue their own banknotes.  English and Scottish banknotes are perfectly acceptable in Northern Ireland, but you may have difficulty using Northern Irish notes in Britain.  Some shops in Northern Ireland will accept Euros. Some will give a good exchange rate but others will not, so check before you pay. 

Credit and Debit Cards

Credit Cards: 
Visa, Mastercard/Eurocard are very widely accepted in Ireland. All hotels, petrol stations (gas stations), supermarkets, shops/stores, pharmacies, restaurants, transport operators, payphones, etc will accept them.
Other cards: American Express (AmEx) is accepted, but not nearly as readily as in the United States or Canada. Diners Club, JCB, Discover etc are generally not accepted. It is not advisable to rely on these cards as a sole method of payment while in Ireland (or much of Europe). 
Debit Cards: 
Irish Laser Cards, Maestro, Visa Debit and MasterCard debit are widely accepted. Other debit cards, not carrying the Maestro, Visa Debit or MasterCard logos are not accepted. Check with your card issuer that your card is acceptable abroad before you travel.
NB: Non-Irish issued Maestro cards are generally not accepted on-line or over the phone in Ireland, but they can be used in retail outlets and ATMs where the PIN is entered. 
Some very small retailers and cafés may only accept cash payment.  Certain retailers will also refuse credit cards for small amounts e.g. less than 10 euro. However, they will often accept Irish debit cards. Some discount supermarket chains, notably Aldi and Lidl only accept debit cards (Laser or Visa Debit). So, do not be alarmed if everyone else is paying by card and you are refused. This may be because their acquiring bank charges them higher fees on credit cards. For this reason, it is advisable to carry some cash, particularly if you plan to shop in small stores or are in very rural areas. Some travelers have found that some B&B are listed in guides as accepting credit cards but may refuse to take them. 
In common with most European countries, over the past few years Irish credit and debit cards have become 'chip and pin'.  If you have a 'chip and pin' card  you no longer sign but key in your pin into a pinpad.
If your card does not have a chip, don't worry as your card will usually still be accepted and you can just sign for your purchase. However, you may be asked for ID for larger transactions. There have been reports that some retailers may refuse non-chip cards, if this happens, ask to see a manager.
They may be able to authorize the transaction manually when they check your ID as they are liable for losses should your card be fake. Also please note that some automated ticket vending machines no longer accept non-chip cards. Automatic checkouts in some supermarkets also no longer have facilities to take magnetic cards. However, staff will be able to process your card manually on a separate terminal.
A very small minority of retailers have inflexible, blanket policies on non-chip cards and will refuse them as they will not accept the liability for losses in the event of fraud. In this case you may need to use an ATM to withdraw cash. If this happens, please do not take it out on the counter staff as they may not be allowed to accept non-chip cards, or their terminal may even be configured to prevent them from doing so.
In general, if you are travelling to Europe, it is advisable to ask your bank if they can issue a chip and pin card. Some US banks have recently begun the process of migrating to this technology too. So, in a few years this should be a non-issue.
If you are from a country outside the euro-zone, when paying with your credit card be sure that you are being charged in euro, and not your own currency.  Some outlets, particularly those popular with foreign visitors, use a facility called 'Dynamic Currency Conversion' (DCC) so that they can apply the charge in dollars, pounds, yen, etc.  However, the rate they use to convert your purchase is invariably worse than that which would be applied by your own credit card issuer, so you lose out.

ATMs 

Almost all ATMs in Ireland provide credit card withdrawals for Visa and MasterCard credit cards and ATM cards with the Cirrus or Plus logo.  Amex and Link cards are also accepted at some ATMs, including AIB and Bank of Ireland (the two largest banks).
ATMs are usually located outside bank branches, but increasingly can be found in convenience stores, supermarkets, petrol stations and even bars. In Ireland there are no additional charges for using in-store ATMs. Charges for using the ATM are dependent on the Bank of the cardholder so check before you leave home. In general the best rates can be found using your own ATM card, as long as your bank does not charge ridiculous fees.
Before you travel to Ireland, check with your bank that your Credit Card, Debit Card or ATM card is activated for use in both the Republic of Ireland and Northern Ireland (United Kingdom). Otherwise, you may have problems withdrawing cash.
All Irish issued cards have 4-digit PINs but ATMs are programmed to accept PINs of different formats for non-Irish cards.  
Most ATMs will recognize that you have inserted a foreign card and offer you a range of language options. 

Travelers Checks

Travelers checks are not widely accepted for purchases and you will generally have to exchange them at a bank or Bureau de Change for euro cash.  This may mean that you are paying commissions twice - once to buy then to sell.  Do your sums before you decide to use them.

Changing Money 

Banks all have currency exchange desks (for exchanging cash, cashing travelers checks, credit card advances etc) and are usually open from 10-4 every day.  One day per week banks are open until 5pm - in Dublin this is on Thursday. Permanent TSB branches are open every day till 5pm.  Banks will apply a handling charge for exchanging cash or check - this will be displayed on a board along with the exchange rate.  Rates and commission charges do vary so it can pay to shop around. Exchange rates can be compared online using sites such as 
Some department stores also provide foreign exchange services but post offices no longer provide it. 
Some independent bureau de change are open longer hours than the banks.  The bureau de change at the airports keep long hours to suit flight times- check the airports own sites e.g. Dublin ,  Cork  and Shannon for exact hours and locations.   Most major hotels will also carry out the same facilities, although rates may not be as attractive.  Note that for security reasons, US$  100 bills are not readily accepted, even in banks.  

Currency Controls 

There are no exchange controls  in Ireland. Any sum of money in any currency can be freely brought  into or taken out of the country without disclosure or other  formalities.

Taxes

In Ireland VAT ("Value Added Tax") is applied to most goods and services.  The rate depends on the type of product but most items are subject to a rate of 21%.  The only positive part is that VAT is included in the price marked on the item so unlike in North America you do not have to do complex mental arithmetic at the checkout !  The 21% rate is higher than most other countries and there are also very high duties on alcohol and tobacco products.  As a result prices in Ireland are often more expensive than elsewhere. 
There is a scheme whereby visitors from outside the European Union can purchase goods to take home with them and benefit from tax relief. Relief is not allowed on services (for instance, hotel accommodation, meals, car hire etc.).  For more information see the Irish Revenue Commissioners website or one of the companies which process the refunds such as  TaxBack.com