イギリス is used to refer to the United Kingdom which consists of four main countries, England, Scotland, Wales and Northern Ireland. It is also sometimes referred to as 英国 which used more for official names (such as the Embassy). It can be seen alongside similar names such as 米国. The character also gets used in newspapers/convenient abbreviations (eg 日英同盟 is the expression often used to refer to the Anglo Japanese Naval Treaty).
イングランド refers to England. it is used as when referring to the country/institutions designated at that level such as the national football team.
The origin of the use of 英 (and ultimately イギリス）dates back to the name given to British people by Chinese. It was pronounced YING and used phonetically to replicate the Portuguese name of Ingles. (This character's meaning of "great" or "superior" and the name "Great Britain" is coincidence. For the record, Great Britain is the term used to refer to the largest island of "the British Isles" which is a geographical name for the islands occupied by the UK and the Republic of Ireland. "Great" is used in the same sense that people refer to "Greater China", to include Hong Kong and Taiwan, or Greater London.) This must date back more than 300 years, before the union of Scotland with England and Wales so the confusing (and for some, annoying) tendancy to refer Britain/the UK and its people as England/English goes back such a long way it would even seem to have been ingrained into Asian languages several hundred years ago.