The term “domain name” is a very general term and is often preceded by another term to more specifically define to what someone is referring.
First of all, there are different levels of domain names as the following examples and definitions show.

1) Top Level Domains (sometimes called a TLD) (http://www.domain.com)
2) Second level domains (http://www.domain.com)
3) Third level domains (http://seo.domain.com)
4) Country code Top Level Domain (called a ccTLD)

1) There are several Top Level Domain's (TLD's) in existence today, the most widely used is .com. The 1980's brought the existence of several gTLD's (generic TLD's) which include: com, .net, .org, .edu, .gov, .int, and .mil. The public was allowed to register second level domains in only .com, .net, and .org and the others had certain restrictions which needed to be met in order to register a second level domain (e.g., you needed to work for the government in order to register in the .gov TLD).

Other gTLD's came into existence circa 2001, including: .info, .biz, .name, and .pro. Only .info is unrestricted and allows the public to register without owning a business for .biz and being some sort of professional (e.g., doctor or lawyer) for .pro. It appears that new gTLD's will be added in the future. Adding new gTLD's gives people the opportunity to register a domain name in a new TLD that is taken in an existing TLD (e.g., www.candy.com is taken but someone can now register www.candy.info).

There are also several existing sTLD's (sponsored TLD's) which include aero, .coop, and .museum. It looks as if more sTLD's will be added in the future. A sTLD has a “sponsor” which represents a niche community (e.g., .aero represents airports, air pilots, and other air related web sites). http://www.domainbank.com is the largest .aero registrar as of this writing.
You can visit http://www.icann.org/tlds/ for more information regarding TLD's

2) Most people refer to the term “domain name” when describing a second level domain name. A second level domain name is the part of the domain that proceeds the TLD (e.g., domain.com). The second level domain name is limited by several factors such as: length of domain name (about 63 characters is the maximum length), until recently only roman characters have been acceptable (numbers and letters), the only other character is the hyphen, which can not be used at the beginning or end of the domain but only within a domain (e.g., doma-in.com is ok, but domain-.com and -domain.com are not). There are currently programs which are adding umlauts and other characters into the acceptable character realm.

3) Third level domains like seo.domain.com are widespread but used much less that second level domain names. The owner of a second level domain can add infinite third level domains to their second level domain name but web masters tend to add file extensions instead. For example a web master would add domain.com/seo.html rather than seo.domain.com. The www in www.domain.com stands for World Wide Web and is a third level domain that is most often a default set by a webmaster (i.e., an internet surfer could type in “www.domain.com” or just “domain.com” and get the exact same web page). Sometimes you will see larger web sites use www1 and www2 to proceed their second level domain name. These are third level domains which typically are directing Internet traffic to different web servers to help balance the load of a lot of Internet traffic.

4) ccTLD's are representative of sovereign countries, for example a .uk TLD is referencing the United Kingdom. There are many ccTLD's each with their own set of rules for who can and can not register a second level domain in their ccTLD. Each also has its own price point and individual set of rules. You can find a good list of ccTLD's at http://www.iana.org/cctld/cctld-whois.htm

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