What is a Domain Name?
A domain name, simply put, is way of assigning a web address to a web server. A web address is translated from a format similar to 'http://www.opacity,ie' into an IP adresss such as 95.69.3.25. A good analogy is a phone book (Remember those!). You look up a name and address in the phone book and you get the number. In this case the name and address is the Domain Name, the phone book is a Domain Name Server (DNS) and the phone number is an IP addresss.
The domain name is the part of a web address without the www or http. e.g syob.net.
What is a top-level Domain Name?
The top-level domain (TLD) name is the part of a domain name after the last period. e.g .com .ie .co.uk .org .biz etc. It is usually related to the type of website or the country that the domain is registered in. Whatever TLD you decide to go with I would always recommend listing a full physical address and phone number on all websites that involve interactions with potential customers. People like to know they can still ring or call a business if things go wrong.
UK Businesses
UK buinesses can register a .co.uk or a .com without much hassle. Unlike a .ie TLD no proof of connection to the UK is required and they are issued on a first come first served basis by Nominet or a Nominet agent. That being said some UK customers may still subconsciously favour a .co.uk site over a .com site as they percieve they are dealing with a UK based business.
From an SEO perspective there is anecdotal evidence that searches on google.co.uk favour sites with a .co.uk TLD while google.com may favour .com addresses. To be safe you should register your business with both a .co.uk and .com domain to future proof availability. NOTE: UK businesses should not use a .org.uk. They are only used by not for profit organisations and charities.
To simplify things:-
1. UK based business selling exclusively to the UK Market now and into the future - .co.uk
2. UK based business selling globally now and into the future - .com
3. UK based business without a crystal ball! - .com and .co.uk
Irish Businesses
A .ie TLD is more expensive and slightly more difficult to register than a .com or .co.uk TLD. .ie domains are registered through the IEDR (Ireland's Domain Registry) usually through an accredited IE registrar. Businesses who try to register their business name e.g. mybusinessname.ie, will find it difficult if they do not have a RBN (Registered Business Number) by being already registered with the Companies Registration Office. In these unusual (and sometimes illegal!) instances businesses should go for a .com domain.
A .com TLD is cheaper and easier to register since there are fewer checks and they are generally granted once the address is not taken. However for a business trading almost exclusively in Ireland I would always recommend registering a .ie TLD for the following reasons:
It gives confidence to possible customers that you are based in Ireland.
It shows 'you are who you say you are' and that the IEDR have been satisfied of this.
Google uses one!
As mentioned above there are no geographical restrictions on registering a .co.uk domain, so if you can afford it and you may trade in the UK in the future, register your business with a .co.uk address as well.
Summary
If you are trading solely within a UK or Irish market then a .co.uk or .ie address in my opinion is the best way to go. For businesses selling globally a .com gives customers the impression of a global reach. However don't regret not registering a TLD in the years to come because it's likely somebody else will!
I have limited this for simplicity to .co.uk, .ie and .com but there are hundreds of TLD's out there today. You can view a recent Top 100 TLD's here and check out ICANN's list to see the stream of new TLD's coming online.