As you know, to have a visible web page you need a place to stay. That "place" (web site) we can offer a hosting server (which means hosting of web pages) either for free or by paying a certain amount. In the Hosting section I suggest some free and paid plans where your web hosting, and even detailed instructions on how to register for a free server and the definition of the terms that you have to know to choose with knowledge.
On the other hand, once we have the space for our website, visitors need to write something in their browsers to go to our new website. This is the URL or web address that can be either a domain chosen (and paid) for you, for example www.burjkhalifahtower.com or www.google.com or www.100seotips.com or a subdomain ( The usually free subdomains) that have a slightly longer form. We shall then see how to register a domain and also how to relate a domain with hosting, etc, etc.
Simply the domain name is a website name and hosting is a place where this domain name will stay with all of its contents.
Now, the other question, from where to buy the domain name and hosting package. If you will write in Google, it will show you million of websites offering you the cheapest domain names and hosting package. You can select any of them according to your choice. I most use the Blue host company that is world's famous and good services provider. It offers the cheapest package and unlimited domains and bandwidht. I am giving the link of Blue Host, you can visit this and compare its prices.
According to Wikipedia
A domain name is an identification label that defines a realm of administrative autonomy, authority, or control in the Internet. Domain names are also hostnames that identify Internet Protocol (IP) resources such as web sites. Domain names are formed by the rules and procedures of the Domain Name System (DNS).
Domain names are used in various networking contexts and application-specific naming and addressing purposes. They are organized in subordinate levels (subdomains) of the DNS root domain, which is nameless. The first-level set of domain names are the top-level domains (TLDs), including the generic top-level domains (gTLDs), such as the prominent domains com, net and org, and the country code top-level domains (ccTLDs). Below these top-level domains in the DNS hierarchy are the second-level and third-level domain names that are typically open for reservation by end-users that wish to connect local area networks to the Internet, run web sites, or create other publicly accessible Internet resources. The registration of these domain names is usually administered by domain name registrars who sell their services to the public.
Individual Internet host computers use domain names as host identifiers, or hostnames. Hostnames are the leaf labels in the domain name system usually without further subordinate domain name space. Hostnames appear as a component in Uniform Resource Locators (URLs) for Internet resources such as web sites (e.g., en.wikipedia.org).
Domain names are also used as simple identification labels to indicate ownership or control of a resource. Such examples are the realm identifiers used in the Session Initiation Protocol (SIP), the DomainKeys used to verify DNS domains in e-mail systems, and in many other Uniform Resource Identifiers (URIs).
An important purpose of domain names is to provide easily recognizable and memorizable names to numerically addressed Internet resources. This abstraction allows any resource (e.g., website) to be moved to a different physical location in the address topology of the network, globally or locally in an intranet. Such a move usually requires changing the IP address of a resource and the corresponding translation of this IP address to and from its domain name.
Domain names are often referred to simply as domains and domain name registrants are frequently referred to as domain owners, although domain name registration with a registrar does not confer any legal ownership of the domain name, only an exclusive right of use.
The Internet Corporation for Assigned Names and Numbers (ICANN) manages the top-level development and architecture of the Internet domain name space. It authorizes domain name registrars, through which domain names may be registered and reassigned. The use of domain names in commerce may subject strings in them to trademark law. In 2010, the number of active domains reached 196 million.
You can study more visitng the following wiki link wikipedia