I didn't thought much about it from the movie. Some large conlfict decades ago, which involved clones and Obi-Wan being a general in it. Interesting backstory piece, nothing more.

After reading Timothy Zahn's trilogy (which is fantastic and one of the EU pieces I love most dearly) I thought that it was a galaxy-spanning conflict between the Republic's army of regular soldiers led by the Jedi against another army, likely (Separatist) aliens. This also explaining why the Empire was staffed only by humans.

I even though they were two intertwined conflicts, hence the plural "wars". The first, when the Jedi and the Republic defeated the enemy, and then when the semi-crazed Dark Jedi - clones hastily created by the Jedi in desperation due to their dwindling numbers during the war - rebelled, also leading a large host of growth accelerated cloned soldiers. During the second conflict the entire clone army went crazy (to the Dark Side) and aimlessly rampaged accross the Galaxy.

Palpatine used the chaos and usurped power, creating the Empire and smashing the clones/Dark Jedi. Behind the scenes, though, he had been influencing the Dark Jedi or teaching the Force to mentally unstable clones.