Stannis Baratheon is the new king on the block and the middle child of the Brothers Baratheon. Since Joffrey is not Robert's true son, Stannis is technically the rightful Lord of the Seven Kingdoms, the King of the Andals, and a host of other fancy titles.
Personality-wise, he's a bit of a stern fellow. Maester Cressen, who knew Stannis as a boy, notes that people think of Stannis as "hard" (1.Prologue.80) because that's totally what he is. We'll give Cressen this one, but the maester forgot to mention that his lord is very harsh, too, which might not be the best combination of character traits.
Consider how Stannis lets Cressen go from his position as Dragonstone maester: "You are too ill and too confused to be of use to me, old man. […] Pylos will counsel me henceforth. Already he works with the ravens, since you can no longer climb to the rookery. I will not have you kill yourself in my service'" (1.Prologue.154). Not exactly what we'd call letting him down gently.
But Stannis is also being honest. Cressen has grown old, his job has become too difficult for him to manage, and Stannis doesn't want him to die trying to serve him. That last one, in a way, is Stannis showing his sentimental side. We believe he is fond of the maester and truly doesn't want the man killing himself in his service. He just has a really hard time saying it in a softer tone.
If we had to come up with a nickname for Stannis—we don't, but we will—we'd go with Stickler Stannis because this guy is all about following the rules. To the letter.
We need only consider the story of Davos's dismembered digits to see how true this is. During Robert's Rebellion, Davos saved Stannis and his men from starvation at the Siege of Storm's End. Stannis was grateful enough to knight his savior, but he couldn't forget Davos's years as a smuggler—and so to punish him, Stannis had Davos's left fingers cut off to the joint.
Recalling the incident, Stannis tells Davos: "It was justice" and "[a] good act does not wash out the bad, nor a bad act the good. Each should have its own reward. You were a hero and a smuggler" (43.Davos.78). He kind of has a point, right?
However, this could be the reason why no one seems eager to support Stannis's claim to the throne. First, they are rather fond of their fingers as they are, and second, nobody likes having the rulebook constantly cited in his or her face. As such, most of the lords and knights who should have claimed allegiance to Stannis have thrown their support behind Renly, the guy who doesn't have a fondness for the rules and finger fillets.
But the Seven Kingdoms have always shunned Stannis for his cool demeanor and stickler ways. Cressen remembers that Stannis was always in Robert's shadow as a youth (1.Prologue.134). When Robert became king, he awarded the Baratheon family castle, Storm's End, to Renly, a huge slight since Stannis was next in line (1.Prologue.88). Burn.
Is it any wonder Stannis grew up memorizing the rules? Even with the rules on his side, he's had to fight for everything he's ever gotten.
Despite his love of the rules, Stannis begins to cut corners in the morality department to secure his crown.
Morality first takes a backseat when it comes to removing Renly. The younger Baratheon certainly slighted his duty by not supporting Stannis's claim. But stabbing him in the back by way of magical assassin? That is definitely a no-go for Stannis's moral code. The ethics of the Seven Kingdoms considers kingslaying a major taboo, one of the unforgivable sins that the gods are sure to punish.
With that said, Stannis may not be aware of how Renly died. He claims he was asleep when it happened and recalls, "I dream of it sometimes. Of Renly's dying. A green tent, candles, a woman screaming. And blood" (43.Davos.90). This quote certainly makes it ambiguous as to whether or not Stannis gave the order.
But he is definitely aware that Melisandre enters Storm's End to assassinate Ser Cortnay Penrose. When Davos protests that there must be another way, a cleaner way, Stannis balks that he must have Edric Storm (43.Davos.129-130).
So we can see that Stannis's moral ground has gotten shaky, even if it has not completely collapsed beneath him. By his own admission he believes that good acts do not wash away bad ones—so we wonder what rewards Stannis will reap for his acts at Storm's End in future volumes.