The United Federation of Planets, the fictional space government present within the Star Trek universe, was designed to show a utopia of human creativity and ingenuity, a fictional representation of humanity's future. While the franchise plays with this idealized image in the later series (especially during Picard) it is often shown to be the pinnacle of a united Government. A large part of this is Starfleet, an organization run by the Federation — but what exactly is it?
Starfleet actually predated the formation of the Federation. It was originally created specifically by the united Earth government, specifically the United Earth Space Probe Agency, or UESPA for short — essentially, it was a more practical NASA. Things get a little hazy when regarding when exactly it was founded, though its origins are thought to be somewhere between 2112 and 2136. Around this time it funded projects by Zefram Cochrane, who went on to discoverer warp capabilities and initiate first contact with the logic fanatics, the ever-so-stoic Vulcans. Once the Federation formed in 2161, Starfleet gradually shifted from being a human-run, specifically UESPA, organization, to being one supported and maintained by, and thus serving, the Federation as a whole.
While it might be easy to classify Starfleet as the Federation's military, that definition wouldn't be quite right. Fans are divided on this definition, as Starfleet has numerous parallels with modern military organizations, but these parallels also stir a lot of conflicting emotions. They use the same military ranks, such as commander and captain, follow the same military chain of command, and even have very similar regulations and terminology. On top of this, someone who breaks the rules set by Starfleet is court martialed, a term used multiple times throughout the franchise, which literally means "a judicial court for trying members of the armed services and military." If this was not enough to draw striking parallels, each vessel within the fleet is armed, even the science ships and shuttles. And whenever there is any cause for a combat situation, like a mass scale invasion from the Borg or a war with the Klingons, Starfleet is the one to respond.
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The reason classifying Starfleet as a military organization doesn't sit well with fans is that, while in essence it is military grounded, it embodies and represents a lot more. Starfleet was founded to defend the Federation, not just to act as the arm that beats its enemies. A lot of people think of the military as a device used by governments in times of war, or to intimidate or defend a populous by force. However, this is a very modern view on the military. Starfleet is much more like Western militaries of the 18th and 19th centuries, specifically the navy. Back then, the British Royal Navy conducted trade missions, partook in diplomacy and, most importantly, conducted missions of exploration. It’s no coincidence that the names of the starships sound familiar: the SS Beagle, USS Challenger, USS Endeavour, and of course, the USS Enterprise. All these were also Royal Navy ships that were set apart from the warships. These vessels were designed specifically for explorations rather than battle, while still armed like one just in case of hostile interactions. It was a time when the world was a much bigger place, and these embassies had to act on behalf of their government that was too far away. This is much like what space becomes for a culture of space-faring people.
It’s interesting to note here that, while modern-day Earth has been explored to the point where there are very few mysteries left to discover, the modern military still supports large-scale scientific explorations in the few areas that are still shrouded in some form of mystery. The frozen wilds of the Antarctic are one example, but so is an effort to map out the ocean floor, a place almost as alien to us as actual space itself. Humanity has only successfully mapped a staggeringly small 5% of it — who know what could be lurking down there. It could be argued that if more of the world was undiscovered, modern-day military might look and act a lot more like Starfleet.
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It is far to say then that Starfleet is military, but is more akin to the navy than the army. Even today, the navy shares a lot of similarities to that of Starfleet, as many navies engage in not only milliary missions, but also in emergency relief and humanitarian aid, protecting from modern-day pirates and assisting various cultures around the world. If anything, when looking specifically at the modern US military (arguably the most common comparison to Starfleet) it's fair to say that their involvement with diplomacy and engagement with foreign counties outweighs actual combat missions.
Starfleet adds to this model, however, emphasizing a massive scientific exploration element. And of course, it incorporates the directive of first contact. These elements cannot be compared to modern military organizations. The biggest differentiation here is that, while they are a military force capable of great destruction, it is not militaristic in nature like the Klingons, for example. Starfleet is designed to be an embassy of peace. They're never looking for a fight, but rather exploring the universe, and protecting it when necessary. Their main directive, after all, is to “seek out new life and new civilizations” not destroy them.