Humanity's got a bright future ahead of it...
There were some teething problems, granted. All those wars on the homeworld before someone got around to knocking some heads and making everyone shut up and play along. Still, a unified Earth eventually happened, and you know once there was nobody left to fight with and not much left to explore, the species would go off looking for more trouble. Colonies on other worlds were founded, space travel developed at a rapid pace... and sure enough, there was plenty of trouble out among the stars. Not many alien species decided to come in peace, for some reason. Humanity endured, though, and expanded. Earth and its colonies grew to become the Federation.
There were still some teething problems. Some hotheads decided to go and burn Earth to a cinder in protest, kick off a minor civil war in Federation space, which at the time was about a tenth of the galaxy... Humanity endured, though, and grew stronger. Some alien races willingly came to join the Federation. Others came over after enough bombs had been dropped on them. It wasn't just a human Federation any more, but there wasn't much doubt who was in charge of the show.
Of course, it's not all clear sailing from here on. Every so often a colony will try to break off - some even succeed for a time. There's pirates aplenty, and there's still two thirds of the galaxy outside of the Federation's sphere of influence. Plenty of work for the Federation Navy and the masses of ground troops it keeps carting around.
And plenty of opportunities to get rich on the frontiers of "civilised" space.
(That's the bare bones of the world setting. If you need further help setting up anything, or further questions about the setting for purposes of character background, ask away.)
A brief timeline:
2115 - First human colony outside of the Solar System
2234 - Razing of Earth
2805 - Current year
A brief listing of Federation armed forces that player characters may have served in, come in contact with, or heard of:
Army - Your regular ground-pounders, bringing heavy armour and plenty of infantry to any planetside brawl. They rely on the Fleets to get them from fight to fight, but once on the ground tend to work independantly of naval assets. The most numerous branch, but not the most well equipped.
Orbital Drop Troops - While technically part of the Army, they deserve their own entry. Issued with advanced power armour and heavy weapons, these soldiers generally spearhead any invasion. They have a high turnover, though it still comes in second to what the assault infantry units go through.
Special Forces - The most elite fighters among the "regular" armed forces, they tend to work behind enemy lines and can pick their gear from just about the whole Federation arsenal.
Legion - The Federation's ultimate ground force, its safeguard and loyalty control. Superb training, unswerving discipline, and equipment far beyond what regular troops have access to. They have far-reaching political power in addition to military might, and if they're seen out and about anywhere, it's usually a sign that something nasty is about to happen. Any emergent psionic talents are also sure to attract their attention.
Navy - The Federation requires a vast number of ships to maintain its holdings across the stars. The Navy itself is split into a number of Fleets with specific tasks. The First is dedicated to the defence of the Federation's capital. The Second is the Legion's own fleet, and as such has access to the most advanced vessels and weapons. The Third is the expeditionary fleet, ferrying troops from warzone to warzone, as well as battling enemy navies - this is the largest of the Fleets, and usually the most lively assignment. Further fleets are dedicated to the protection of specific sectors.
Travel in the Federation:
Travel is in most cases restricted only by the budget of the traveller, but interstellar voyages are a more complicated matter. While most Navy ships (certainly all capital ships, including carriers and troop haulers) are fitted with their own jump drives to allow them fast and reliable travel between distant stars, most civilian craft have to rely on the Federation-run system of jump gates and naturally occuring warp points to get around. While the jump gates are fast and reliable (unless you get stuck in a long queue), they do allow the Federation to monitor who goes where, which is something certain parties try to avoid. The power requirements of an individual jump drive generally mean that it can only be fitted on larger vessels.
Aliens in the Federation:
There's plenty of life in the galaxy that isn't human. It's varied, but not as much as one might expect - certain evolutionary paths lend themselves to spacefaring more than others. A number of alien races live under the Federation's rule, and though officially there is no discrimination, more often than not such aliens never really rise above second class citizen status. The exception to this rule is the armed forces - loyal service can guarantee numerous benefits for an alien, though within the army aliens generally keep to their own species groups, and mixed-race units are rare at best.