The story of cigarettes being exchanged and a football being kicked around
"Gradually firing ceased almost Everywhere along the line that Christmas Eve. Men got up and sat on their parapets and shouted greetings across the the'enemy'.....The Christmas spirit had simply conquered the battlefield.
When dawn came the next morning , the ground was frozen solid. In some areas a sprinkling of fresh snow lay on the ground... The sudden change in the weather brought astonishment and cheer...
Soldiers moved into no man's land, or in some cases even into each other's trenches, and celebrated. Some were shy. Others were more open They talked, sang. and exchanged stories  and gifts. As the morning wore on, confidence grew. Burial parties were arranged. The 6th Gordon Highanders and the 15th Infantry Regiment, a Westphalian unit, joined in a moving ceremony for the dead. As Scotsmen, Englishmen, Saxons and Westphalians lined up on both sides of a communal  mass grave, the Reverend J Esslemont Adams, minister of the West United Free Church, Aberdeen, and chaplain of the 6th Gordons, read the Twenty-third Psalm in English. A theology student then read it in German: 'Der Herr ist mein Hirt: mir wird nichts mangeln. Er weidet mich aug einer fruner Aue: unde fuhret mich zum frischen Wasser...'
The Lord's Prayer followed, sentence by sentence, in both languages: "Our Father Who art in Heaven. Unser Vater in dem Himmel...'"  Modris Eksteins op. cit. pp 110-11.

Frederic Villiers' depiction  of the 1914 Christmas truce published on the front page of the Illustrated London News on 9 January 1915. The original caption was "The light of Peace in the trenches on Christmas Eve: A German soldier opens the spontaneous truce by approaching the British lines with a small Christmas tree."
https://bshistorian.files.wordpress.com/2013/12/christmas-truce.jpg
https://www.iwm.org.uk/history/the-real-story-of-the-christmas-truce; http://www.bbc.co.uk/guides/zxsfyrd; https://skeptoid.com/episodes/4498. . 
The evidence from individual soldiers letters:
https://off-guardian.org/2017/12/25/oh-what-a-lovely-war-christmas-truce-1914;  https://gm1914.wordpress.com/2016/12/22/christmas-in-the-trenches; https://www.iwm.org.uk/history/voices-of-the-first-world-war-the-christmas-truce
For a comprehensive account: Malcolm Brown & Shirley Seaton Christmas Truce (1994). Based on what Captain R J Armes from the 1/North Staffs regiment mentioned in an excited letter to his wife, it's evident that the temporary truce " was seen as a suitable time for the decent disposal of the dead bodies lying in No Man's Land" (p 69).

Contrary experience https://www.telegraph.co.uk/history/world-war-one/11307513/Christmas-truce-of-1914-was-broken-when-German-snipers-killed-two-British-soldiers.html.
Spontaneous and not-to-be-repeated? This seems to have been the general policy of officers, but further evidence now indicates its happening in subsequent years and on different fronts.
https://www.huffingtonpost.co.uk/entry/christmas-truce_uk_5858f9a2e4b0e9baa8784d32?guccounter=1&guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAANkbEcCbmxlNk9fJ2zaAwU9f7ab7el-IcJ2n0rnZ5cyvdX75Le8WibB-Ls0tL-1OKaz6M-tJ1gFp2g1wJqho-nli1cp7xNlIuWPBhqsT65-R5tHRBdjoIBFQIUOearlzdZEKNvmQvcqlCuhgCAkFzuPBPOUk1Q07zAumwjg7EwHW