This is maybe several questions in one:
It might be a plausible argument to say that the period 1815-1848 was a period of peace between great powers maintained by a conscious effort of balance-of-power politics. But after 1848 that argument is MUCH harder to make. The most that can be said is you didn't have any coalition wars with multiple great powers teaming up — à la the Seven Years War, Napoleonic Wars, and the 20th Century World Wars.
But the decades after 1848 saw (non-exhaustively):
The years after the 1870-1 Franco-Prussian War — the so-called belle époque — saw fewer conflicts pitting great powers against each other, but it was hardly peaceful, with plenty of bloody colonial conflicts, numerous wars connected to the slow disintegration of the Ottoman Empire, a number of inter-state wars in South America, and many civil wars, as well as the Russo-Japanese War that announced the entry to Japan onto the world stage as a power.
At most, if you want to periodicize the 1815-1914 period based on great power conflict, you've got three key periods, not one: a relatively peaceful 1815-1848, a violent 1848-1871, and a more peaceful 1871-1914. And of course even with that you can always point to exceptions, challenge the focus on just "great power war" given how violent conflicts elsewhere could be, etc.
If you're specifically focused on the absence of war due to a formalized commitment to balance of power, then you've got to focus on the 1815-1848 "Congress System" following the Congress of Vienna. That's where you had a conscious collusion by great powers to uphold the status quo, with periodic congresses being held to settle various crises. Specifically, the Congress of Vienna created the "Quadruple Alliance" of Britain, Austria, Prussia and Russia to uphold the terms of the Congress, prevent another resurgence of revolution or Bonapartism in France, and — crucially — to meet regularly in follow-up congresses to resolve later crises. (In 1818, Bourbon France was permitted to join what then became the "Quintuple Alliance.") There was also — especially among the so-called "Holy Alliance" of Prussia, Austria and Russia — an explicit idea that great powers could intervene in other countries to suppress revolution (the "Troppau Protocol").
This, too, wasn't perfect; the "pure" Congress System as set up in 1815 had collapsed by the mid-1820s, but a system of more ad-hoc conferences resolved future crisis until everything fell apart with the Revolutions of 1848. For example, the 1830 London Conference resolved tensions surrounding the Belgian Revolution and the aftermath of France's July Revolution; the 1840 Convention of London resolved another crisis involving Egypt, the Ottoman Empire, and a potential French intervention.
The post-1871 period saw a return to international conferences as a means of resolving disputes, including two congresses in Berlin to address tensions in the Balkans and Africa. But this continued to be on an ad hoc basis, rather than a formal alliance like the immediate post-Napoleonic period.
Again: this is not a black-and-white matter. One could easily make counter-arguments to everything I've said here, including extending the Congress System through to the 1856 Congress of Paris that ended the Crimean War. My own historical judgment is that there's not a single diplomatic throughline that you can draw from Waterloo to World War I, but rather — at a minimum — several distinct periods.
A good source for further reading is Richard Evans' The Pursuit of Power.