Davos Seaworth is basically the Han Solo of Martin's world. He was once the most notorious and elusive smuggler in the Seven Kingdoms (1.Prologue.59), then he joined the rebellion—Robert's Rebellion, not the Rebel Alliance—and became a respectable member of society. Although he knows every trick in the book, he's an honest guy at heart. Still, if it came down to it, we're positive he would shoot Greedo first, too. Just saying.
When we first meet him, Davos is already a knight, but he was born in the Flea Bottom district of King's Landing. During Robert's Rebellion, Davos smuggled onions and salt fish into Storm's End while it was under siege, saving Stannis and his men from prolonged, agonizing deaths. Stannis later knighted Davos for his bravery, but also cut off his left-hand fingers at the joint as punishment for his smuggling days. Whether to remind himself of his low origins, or perhaps because he is proud of them, his sigil is a black ship with an onion displayed on its sail (1.Prologue.73-4).
Despite Davos's obvious bravery, he remains an outsider among the other lords and knights:
Davos would have given much to know what he was thinking, but one such as [Lord] Velaryon would never confide in him. The Lord of the Tides was of the blood of ancient Valyria, and his House had thrice provided brides for Targaryen princes; Davos Seaworth stank of fish and onions. It was the same with the other lordlings. He could trust none of them, nor would they ever include him in their private councils. (11.Davos.12)
This is an important fact when considering Seven Kingdoms' society, and it is perfectly embodied in Davos's experience. Modern Western society considers crashing through the glass ceiling to be a mark of excellence, of succeeding in the world, but in the Seven Kingdoms, the social stratum is much more ridged. People are born where they belong and should work hard to be the best their birth allowed, whether knight or baker or lordling. After all, if you were supposed to be a knight, you would have been born into a knight's family. Duh.
This is why Davos is scorned in the quote above and why he tells Cressen, "To them I'll always be the Onion Knight" (1.Prologue.65). By moving up in the world, Davos has disrupted the social structure. His becoming a knight suggests that anyone can do it (well, anyone brave enough at least). And if anyone can do it, then is it really that special to be a knight?
The same is true of Davos's sons, but Davos consoles him by reminding himself, "My grandsons will joust with theirs though, and one day their blood may wed with mine. In time my little black ship will fly as high as Velaryon's seahorse or Celtigar's red crab" (11.Davos.12). Assuming Stannis wins the Iron Throne, of course.
We wouldn't exactly say that Stannis and Davos are BFFs. Instead, Stannis sees Davos as a valuable and loyal knight because, well, he is. Davos is also honest with Stannis when the other lords are too busy kissing up, making Stannis remark, "He makes me wish I had more smugglers in my service" (43.Davos.103).
But why is Davos so loyal to Stannis? That answer is quite simple: Davos believes, "Everything I am, I owe to Stannis" (11.Davos.14). Since he owes his lord everything, he follows him loyally. As for the finger incident, Davos believes that the move was just since he did disobey the king's law and bad deeds deserve punishment.
Still, though, there are signs of strain on their relationship. When Stannis tells Davos he plans to assassinate Ser Penrose to take Storm's End, Davos protests, "My liege, you must have the castle, I see that now, but surely there are other ways. Cleaner ways" (43.Davos.129). Davos believes that Melisandre has begun to change the lord he once swore his loyalty to, but he goes along with the assassination plan because of his vows. It totally makes him sad, though (43.Davos.133).
Davos spent years smuggling, and as a result, he has a pretty realistic outlook on life, liberty, and the pursuit of happiness. Honor? Reputation? The singing of songs to account your glory? Eh, those are all well and good, but Davos would rather survive to live another day.
His realist nature has led Davos to become a pretty decent strategist despite his lack of formal training. When Stannis insists he must take Storm's End, Davos advises that the castle is best abandoned. He argues that only the Lannisters can harm them, and once they take King's Landing, Storm's End will be theirs anyway (43.Davos.104). Davos doesn't care whether they have Edric Storm to prove Joffrey's dubious parentage; he just wants to get the job done with as little danger to their forces as possible.
We see another example of Davos's pragmatism during the assault on King's Landing. He thinks it would be best to send "the swiftest ships to probe upriver and see what await[s] them, instead of smashing in headlong" (59.Davos.10). Seems reasonable enough to us. Ser Imry, the Lord High Captain, ignores this obviously good idea, though. Why? You guessed it: because he sees Davos as a lowborn craven.
But Davos doesn't care about what people think of him. Lowborn? Sure thing. Coward? If you insist. What matters most is doing the job well and doing it clean. You got to respect that.