What are godchildren/godparents?

Besides the more ritual and\or material functions, godparenthood implicitly included the acceptance of the responsibility to provide for minor godchildren.

A godparent (also known as a sponsor, or gossiprede), in many denominations of Christianity, is someone who bears witness to a child's christening and later is willing to help in their catechesis, as well as their lifelong spiritual formation.

Alexander, Crown Prince of Yugoslavia (born 1945)

Queen Elizabeth became the godmother of Alexander, Crown Prince of Yugoslavia back when she was still Princess Elizabeth. Elizabeth's father, King George VI, was also his godparent

Charles Strachey, 4th Baron O'Hagan (born 1945)

Elizabeth became the godparent of Charles when she was still Princess Elizabeth. Born in 1945, Charles attended Eton College and served as a Page Honour to the Queen between 1959 and 1962.

Julian Hardinge, 4th Baron Hardinge of Penshurst (born 1945)

Elizabeth's godchild Julian has stayed out of the public eye throughout his life.

Elizabeth Lavinia Sara King (born 1946)

Elizabeth Lavinia King typically goes by her middle name, Lavinia. She's the daughter of Lieutenant Commander James Osborne King and the Honorable Elizabeth Patricia White. The Queen and Elizabeth White were childhood friends.

Rosemary Elizabeth Elphinstone (born 1947)

Rosemary is the daughter of the Queen's former lady-in-waiting, Jean Hambro,

Jean married Andrew Elphinstone, Rosemary's father. Andrew also happens to be the Queen's cousin; his mother, Mary Elphinstone, Lady Elphinstone, was the Queen's maternal aunt

Sandra Elizabeth Zia Butter (born 1948)
Sandra is the daughter of Major Sir David Butter, a descendent of an ancient Scottish family, and Lady Myra Butter, who is a cousin of Prince Philip's and the granddaughter of Grand Duke Michael of Russia. Myra is one of Queen Elizabeth's closest friends from childhood

Edward Philip Gerald Hay (born 1949)

Edward is the son of Sir Alan Philip Hay and Lady Margaret Katherine Seymour. Edward served as Page Honour to the Queen from 1964 and 1966. Edward's mother, Lady Margaret, was a lady-in-waiting to Elizabeth, and later held the office of Woman of the Bedchamber from 1953 through 1975

The Hon Michael-John Knatchbull (born 1950)
The son of John Ulick Knatchbull, 7th Baron Brabourne and Lady Patricia Mountbatten, Michael-John is related to the royal family. His mother, who was also known as Countess Mountbatten of Burma, was Prince Philip's first cousin

David Lascelles, 8th Earl of Harewood (born 1950)

David is a great-grandson of King George V and a first cousin, once removed, of his godmother, Queen Elizabeth. His father, George Lascelles, 7th Earl of Harewood, was the son of Princess Mary, Princess Royal, who was the only daughter of King George V and Queen Mary.

Caroline Elizabeth Longman (born 1951)

Caroline's mother, Lady Elizabeth Longman, was a childhood friend of Queen Elizabeth's and a bridesmaid at her wedding.

Elizabeth Georgina Gail Villiers (born 1952)

Known by her middle name, Georgina is a daughter of Nicholas Villiers, whose father was George Villiers, 6th Earl of Clarendon

Elizabeth Harriet Colville (born 1952)

Elizabeth Harriet Colville goes by Harriet. Her mother, Lady Margaret Egerton, was a lady-in-waiting to Queen Elizabeth and to the Queen Mother.

In 1976, she married David James Bowes-Lyon, and they have three children.

Victoria Ann Rhodes (born 1953)

She is the daughter of Margaret Rhodes, whose mother is Lady Mary Bowes-Lyon, the sister of Elizabeth's mother, the Queen Mother. Margaret and the Queen are first maternal cousins

The Lady Virginia Mary Elizabeth FitzRoy (born 1954)

Lady Virginia FitzRoy is the daughter of Hugh FitzRoy and Ann Fortune FitzRoy, Duchess of Grafton. Ann currently serves as the Mistress of the Robes within the Royal Household, a position she has held since 1967.

Christopher Abel Smith (born 1954)

Christopher is the son of the Queen's former lady-in-waiting, Henriette, Lady Abel Smith and Brigadier Sir Alexander Abel Smith. 

Princess Friederike of Hanover (born 1954)

Princess Friederike is the daughter of Prince George William of Hanover and Princess Sophie of Greece and Denmark, Prince Philip's sister, making Queen Elizabeth both aunt and godmother to Frederike

George Herbert, 8th Earl of Carnarvon (born 1956)

Herbert is best known for being the longtime resident of Highclere Castle, where Downton Abbey was famously filmed. 

Alice Dugdale (born 1957)

Alice's mother, Kathryn Stanley, also known as Lady Dugdale, was one of Queen Elizabeth's longest-serving ladies-in-waiting

Fergus Leveson-Gower, 6th Earl of Granville (born 1959)

Katherine Elizabeth Somervell (born 1961)

Katherine is the daughter of Diana Cinderella Mildred Bowes-Lyon and Peter Somervell and is related to her godmother, Queen Elizabeth: her mother, Diana Bowes-Lyon, is the daughter of John Bowes-Lyon, who was Queen Elizabeth's maternal uncle. 

James Arthur Hussey (born 1961)

Hussey's mother, Lady Susan Hussey, served as one of Queen Elizabeth's ladies-in-waiting for over 50 years

David Armstrong-Jones, 2nd Earl of Snowdon (born 1961)

David Armstrong-Jones is the son of Queen Elizabeth's sister, Princess Margaret, making the Queen his aunt and godmother.

Edwina Victoria Louise Hicks (born 1961)

Edwina, who now goes by her married name Edwina Brudenell, is the elder daughter of Lady Pamela Hicks and David Nightingale Hicks.

Lady Pamela, is a first cousin of Prince Philip.

James Robert Bruce Ogilvy (born 1964)

James Ogilvy is the son of Princess Alexandra of Kent and Sir Angus Ogilvy.   Princess Alexandra is the Queen's paternal cousin.

The Hon. Charles Edward Maurice Spencer, 9th Earl Spencer (born 1964)

He is first and foremost Princess Diana's brother, as well as being the Queen's godson.

Princess Theodora of Greece and Denmark (born 1983)

Theodora is the daughter of deposed Greek King Constantine and Queen Anne-Marie and a 4 x great-grandchild of Queen Victoria.