Correction, flag of the United Kingdom. The flag of Great Britain doesn't have the St Patrick cross as Ireland was not fully incorporated under the crown until 1807 I.e the formation of the United Kingdom of Great Britain and Ireland. Wales is not featured because it's a principality, not a kingdom.
The Union Jack is the national flag of the United Kingom and Northern Ireland. The design of the Union Jack combines aspects of three older national flags: the red cross of St George of England, the white saltire of St Andrew for Scotland and the red saltire of St Patrick to represent Ireland. Wales, despite also being part of the Union, has never been represented on the flag.