Solars are the Chosen of the Unconquered Sun. Their overall concept is pretty straightforward: they are simply very, very, very good at what they do. Most people who end up becoming Solars are already naturally talented, skilled or accomplished in some manner, and probably would have gone on to do some great things (for good or ill) in their lives even before Exaltation. Or they are people who have immense potential to become great, if only they had the opportunity.
Solars usually receive their Exaltation when facing an insurmountable challenge and choosing to push through anyway. Often, this means they Exalt during a moment of crisis and only survive the ordeal because they Exalted. Since it comes with no warning and no explanation (except for Zeniths, kinda; see below), most of the time you'll have no idea what just happened besides wow, you have a lot of power now.
There are five Castes (archetypes) of Solars, corresponding to different aspects of the Unconquered Sun and the roles they played in ancient society.
Dawn: The peerless fighters, champions and generals. They aren't always warriors or start out as one, but they are always people who have had battle be a major part of their lives in one way or another. Their special abilities improve their and their allies' combat skills, and they can become so intimidating that they inspire fear even in beings that explicitly don't have emotions.
Zenith: The priest-kings, prophets and charismatic inspirational leaders. Unlike the other castes, Zeniths get a message from the Unconquered Sun himself when they Exalt, urging them to make the world a righteous place (so they actually have an idea what they just got themselves into). Their special abilities are essentially empathetic exorcism, Smite Evil™, and commanding spirits to materialise (on the authority of the Sun).
Twilight: The scholars, crafters and wizards. They're nerds the ones who know all sorts of lore, and are always people who have the pursuit of knowledge as a major theme of their lives. Their special abilities help protect them from harm, such as projecting a protective aura and temporarily poofing themselves out of existence. They can also make lesser spirits their familiars.
Night: The spies, assassins and thieves. Sometimes, you have to fight your enemies on their terms, and there's dirty work to be done. People forget that just because the Sun isn't out, it doesn't mean he isn't watching you in the darkness. Nights' special abilities let them use their other abilities undetected and improve their stealth. They can also temporarily become truly concealed and unidentifiable even by supernatural means.
Eclipse: The diplomats, socialites, merchants and trailblazers. They're typically the people-savvy ones who act as the public face of their group, forging alliances not only between mortal men but also the various supernatural beings of the world. Their special abilities let them sanctify oaths they witness (with Suitably Dramatic Consequences for oathbreakers), grant them diplomatic immunity when dealing with supernatural beings, and let them learn certain abilities from said beings that other people can't.
This will go a bit into the actual gameplay system to try and explain how exactly Solars are 'exceptional' compared to the other Exalt types, and stuff you may want to think about when building a Solar, so bear with me a little.
When building a Solar, you pick the Abilities (basically, skill groups/trees) that your character has. Each Solar Caste has a selection of Abilities associated with it that complement the archetypical roles of the Caste. You pick 5 from that selection as your character's Caste Abilities (and further customize your character with other ability choices after). You can think of Caste Abilities as being part of the reason why your character Exalted as this particular Caste and not some other one. The Caste Abilities for each caste are as follows:
Dawn: Archery, Awareness, Brawl (and Martial Arts), Dodge, Melee, Resistance, Thrown, War
Zenith: Athletics, Integrity, Performance, Lore, Presence, Resistance, Survival, War
Twilight: Bureaucracy, Craft, Integrity, Investigation, Linguistics, Lore, Medicine, Occult
Night: Athletics, Awareness, Dodge, Investigation, Larceny, Ride, Stealth, Socialize
Eclipse: Bureaucracy, Larceny, Linguistics, Occult, Presence, Ride, Sail, Socialize
Then, from the 5 Caste Abilities you picked, you choose 1 of them to be your character's Supernal. Your Solar's Supernal Ability is, basically, their Signature Thing. It's what they are the most exceptional at, and probably what earned them their Exaltation. Gameplay-wise, when learning Charms in your Supernal Ability, you can ignore the Essence (level) requirement as long as you meet the other prerequisites to learn them. This means even baby Solars can potentially have high level Charms that allow them to do ridiculous bullshit and punch above their weight class.
To use an example in Imeeji, Gracious Skylark's (Eclipse's) Supernal is Socialize, which is the Ability that covers certain social skills, how well you apply them in social situations, and how well you can read others. He is ridiculously good at reading people in particular. To the point where he can figure out a person's figurative price ("Every man has his price."), if he tried hard enough. In gameplay, he's still a relatively new Solar (Essence 1), but that particular skill is an Essence 3 Socialize Charm.
Basically, Solars are terrifying because they take talent and skill to wuxia/anime bullshit levels as their main shtick.