When people refer to the different "types" of domain they are usually referring to the domain extension, for example .co.uk and .com. For the majority of the history of the internet these extensions were limited in number and restricted to country specific extensions such as .co.uk and .de and the familiar generic extensions such as .com, .net, .org and the like. 

As of 2014, many more domain extensions are now available such as .london, .gym, guru and so on. Although this allows for more flexibility in terms of marketing the vast majority of domains purchased are still those that have been in existence for years such as .com and .co.uk. 

it’seeze can register any kind of domain for you although certain unusual domains will carry a premium price and will not be eligible as your free domain.