The Alliance is the organistion that tries to control all the planets in the new solar system. Their main sphere of influence is the central planets. They were responsible for the creation of the Reavers after experimenting on people on the planet of Miranda. The Alliance had hoped to create a chemical that would subdue and passify the population, for the vast majority it worked but for others, they became cannabalistic and attacked everything in site.
The Alliance are after the Serenity because onboard is River Tam who in her head is the information about what happened on Miranda and they want to silence her for good otherwise people will rise up against the Alliance. The Alliance send The Operative to bring back river Tam at no expense spared.