I've often spoke out about who is the best princess (vs favorite princess). This isn't going to be about who is my favorite princess (but it isn't wholly unrelated), but rather whom I feel is the best overall princess and why. I'm going to take into account everything I've read and seen across all media for this.
Starting from the bottom... Princess Luna, probably the most popular princess, is also, in all likelihood, the worst princess. To start with she is overly tied to emotions. While this makes for a very engaging character, it makes for a poor leader. It has caused her issues in the past and present. Her emotions are what lead her to be taken by the nightmare force, and try to kill her sister. Furthermore, she nearly cancelled a popular holiday merely because she disliked it. Yes, she helps individual ponies with nightmares, but that does not help the overall masses. She also lacks the experience her sister has (though that is mostly the fault of being banished which was her fault for letting her emotions get in the way of doing her duty for the ponies she guards)
Cadence would probably be next, but it is rather close with Twilight, but that is mainly due to lack of information, so a lot of this will be conjecture. Even in the book explaining her ascension to princesshood, we really don't have much to go on. That said, she has been shown to be at least a somewhat effective leader in the Crystal Empire, putting the well being of her subjects before her own. In fact she nearly passed out trying to protect them from Sombra. While she hasn't had much screen time it is clear she places great value on those around her and will do anything to help, whether it is individual or group.
Twilight would be after... While her time as princess is short, we have seen her attributes countless times over the course of the series. From each of the main big bads from season openers and closers to times like the Winter Wrap Up. She has shown that not only can she lead but she's willing to when it is required of her. Naturally though she is reluctant, and we have yet to see where the "Princess of Friendship" path will go, and we likely won't until Season 5. But still, she has shown that she is a true leader and will put the well being of others before herself. The only reason she exceeds Cadence is due to the plethora of information as the lead character.
And at the very top is Celestia. This isn't even really a contest. From the start she was exalted above all others as a great ruler, all the way back in the first episode. But let's forget hearsay and focus on what we actually know of her. First is that she thinks of everyone, even when it is painful to her. She continued the Summer Sun Celebration as if nothing was wrong despite it being greatly painful for her as it was a reminder that she had to banish her sister. It can be postulated that she had similar feelings about Nightmare Night. Though Nightmare Moon was just a character to many ponies from an old ponytale, to Celestia it was always her sister that went astray. She kept peace and prosperity, alone, for around 1000 years, and was in a position of power long before that in all likelihood. She is also the princess most likely to make decisions and stick by those decisions even among other princesses, even if that decision is tough or possibly controversial (eg Having Discord face off against Tirek, knowing full well that sending anyone else would have likely lead to disaster, despite it being a decision that did not sit well with the other princesses). She may not be seen doing much anymore in the way of actively protecting ponies, but it is clear that she acts as an ambassador personally to other lands, having gone to Saddle Arabia and likely heading the talks with Maredonia. In short Celestia is an actual leader. If something happens in Equestria she is the one everypony runs to, including Luna, Cadence and Twilight. She has the experience, poise, and most importantly she has all the qualities of a great leader, including ability to inspire, confidence, commitment and a positive attitude (not to mention that she has a sense of humor).
While it is easy to see why others prefer one princess over another as a favorite being characters, however the question of best princess is a simple answer. Celestia is clearly the leader of Equestria, other princesses defer to her, her subjects are most likely to come to her for issues concerning Equestria. While it can be argued that the other princesses may be better characters, or at the very least better developed (most of Celestia's characterization is from hearsay rather than from what we actually have seen), it cannot be argued that she is not the true ruler of Equestria, and therefore the best princess.