Kingdom stats (which can be found by clicking the "Stats" tab on the top left while on the kingdom map screen) define the collective health of your realm. There are ten kingdom stats which quantify how developed your kingdom is in various areas: Community, Loyalty, Military, Economy, Relations, Divine, Arcane, Stability, Culture and Espionage.
Barony Stats (1/4): The barony has ten basic stats, each supervised by a Leader. They are as follows: Population (Regent), Loyalty (Councilor), Army (General), Economy (Treasurer), Religion (High Priest), Foreign Affairs (Grand Diplomat), Stability (Warden), Magic (Magister), Culture (Curator) and Espionage (Minister).
You can improve your kingdom stats by constructing buildings, or by successfully completing Event Cards (Opportunities). On the other hand, you'll lose kingdom stats if you fail or neglect Event Cards (Problems), and in particular main quest-related events, which tend to be more impactful and recurring until you resolve said quest. Should any of your kingdom stats be reduced too low, it'll start affecting the level of unrest of your kingdom - should your unrest get out of hand, your kingdom will collapse.
Stat Development (2/4): Stats increase if your governance is successful and decrease if some events don't end well or if negative factors are placed in your way. Don't let your barony stats drop to zero or lower. At this point your subjects will riot, and your state will begin to crumble.
Each stat has certain milestones or ranks within them. Every 20 points of a stat will increase your rank. Rank One unlocks a Leader in a corresponding position. Further development of the stat and upgrade of its ranks is possible only if you have a leader in the relevant position.
Obviously, then, your kingdom stats are fairly important. They're the objective measure of your kingdom, and virtually every aspect of kingdom management (Leaders, Event Cards, construction) are important inasmuch as they affect your kingdom stats.
Fortunately you're not alone in managing your kingdom, as each of your ten kingdom stats has a leader overseeing all things related to that stat. Your Military is overseen by your General, your Economy is the purview of your Treasurer, your Population is governed by your Regent, the Councilor is tasked with the upkeep of Loyalty, the High Priest manages matters of Religion, and so on.
Leaders will be discussed in greater detail, below, but they do pertain to kingdom stats in a variety of ways. First, what Leader you assign to resolve any particular event (many have two possibilities) determines, in part, the kingdom stats you'll gain from completing said event. It's also worth noting the Roman numerals beside each kingdom stat's value - these reflect the rank of that particular kingdom stat. After every twenty points in a kingdom stat, that stat will be eligible to rank up, which can be done by returning to your throne room and resolving an event related to the rank up. After this, you'll need to complete a project, to which you'll typically need to devote your undivided attention (fourteen days will usually be automatically advanced). After this, the kingdom stat the Leader is in charge of will rank up, which generally increases your ability to resolve higher difficulty checks and may unlock new projects that grant you other passive bonuses.