Ameteur here. A fair number of samurai became involved in bureaucracy and administration, but the rest were somewhat listlessly bored. Their boredom eventually saw them contribute a great deal to the growth of Edo culture, although necessarily less than the merchants since their only income was an often insufficient stipend. There was also a strong movement in favour of them reconnecting with their roots that saw ideas like kenjutsu (swordplay during the warring period) not fade away but transition into things like kendo (still swordplay but the 'do' implies more of a focus on artistry than practical usage.). They (not quite in accurately since the closest they came to actually warring was brawls and excecuting 'disrespectful' peasants) saw themselves as having gone soft as a class so idealisations of themselves as following a stiff moral code etc appeared nth in popular culture and books whining about this having gone soft such as 'Hagakure' (a very morbid book about essentially how 'the way of the samurai' is death and battle, which is pretty silly considering there had been no wars in living memory when it was written).

Sources: Kenneth Henshall's a History of Japan Marius Jansen's The Making of Modern Japan Lots of other reading.

Please stop recommending works of fiction, thank you.

To clarify, several users based their entire answers on fictional works. Their answers have been deleted.

After the end of the Sengoku Jidai, the life of a Samurai became more concerned with intellectual pursuits - poetry, calligraphy and so on. These had always been practised by samurai, but in the absence of war they became more dominant. Martial Arts such as Kenjutsu (swordsmanship) were still practised, but over time these became more ritualised and less to do with actual combat. As /u/PwnedDuck has mentioned, many Samurai became involved in administration and bureaucracy and lived as civil servants, since this fulfilled the Samurai value of servitude to one's feudal lord (effectively the Shogun). Philosophy was another common pursuit, especially philosophising about Bushido itself. Many Samurai were keenly aware of the contradiction of being a warrior class without a war to fight, and sought ways to reconcile this.

Another important aspect of life for many Samurai in the Edo Period was Sankin Kotai - the legal requirement to have a residence in Edo, and to spend a certain portion of the year there. The reason behind this was to prevent rebellion by keeping the Daimyo (Samurai Lords) under supervision, but it also bled dry the wealth of the Samurai, since convention required the Samurai to enjoy very lavish lifestyles during their time in Edo.

If you are interested, there is an excellent In Our Time podcast on 'The Samurai,' which delves quite heavily into this question. I am basically just summarising some of its contents here.