In Game of thrones, each house seems to have a common trait among the family, for example:
Does Lannister imply intelligence?
They show Tyrion and Ceresi thinking intelligently and taking intelligent measures
Does Stark mean honour and loyalty?
eg: Eddard and Robb
Does Baratheon mean foolish and incapable?
eg: Stannis and Robert
Does Targaryen mean power and potential?
eg: The Khaleesi
Is that True? or What does each group mean?
Interesting question. I'm not sure how satisfactory the answer will be though. The series (and to a larger extent the books) tend to shy away from black and white characterization. There are however a few common stereotypes among houses. Things that are talked about in-world but aren't necessarily true. Think blonde jokes. Anyway, here are my thoughts on the four major houses:
They have been the Winter Kings since antiquity, until King Torrhen Stark surrendered his crown to Aegon I Targaryen rather than plunge his people into a costly and bloody war. Stereotyped to be honorable to a fault. Their family words "Winter is Coming" has earned them the reputation of a grim and humorless people, who are always on the defensive. The fact that they seem to be the only major house to take the Night's Watch seriously only reinforces that stereotype. Like most Northerners, they also give the impression of being aloof and distant from the day to day political maneuvering so common in the South.
Legend has it that they are descended from Lann the Clever, and this has shaped the public conscience to think of them as clever. Their lands are rich with gold mines which has made them the richest house in Westeros and those riches are also now part of their image. Their unofficial motto "A Lannister Alway Pays His Debts" plays on the image of the clever rich bastard that you don't want to cross. An exception seems to be King Joffrey who seems to be straight as an arrow in regards to what he wants. But he does, however, share their other most known trait: pride. Their sigil of a lion and their official words: "Hear Me Roar!" is indicative of this.
A few traits seem to have followed them throughout the ages. The Baratheons are large, muscular men, and they tend to have a love for war and life in general that they do not care to hide, and are terrible foes when angered and they are angered quickly. "Ours is the Fury!" are their family words. Both Robert and Renly (in the books at least) are large men who dwarfed those around them, and they both had a martial flair and loved laughter, feasts, and tourneys. Ned feared Robert's anger even though he was his closest friend. Stannis, however, seems to be the exception, and it probably explains why he was so distant from his brothers.
The first trait that comes to mind is: foreign and otherworldly. The Targaryens are not native to the land of Westeros. They arrived only about three hundred years before the start of the story, fleeing the doom of their own homeland Valyria. While they have taken up a few of the customs of their conquered people like speaking Westerosi and adapting the Faith of the Seven, they still clung to their Valyrian roots. They continued to at least learn High Valyrian speech, they have for the most part very distinctive coloring (white-blonde hair and purple eyes), they raised dragons, and they (for the most part) married brother to sister to keep the bloodline clean. This tendency to incestuous marriage has made madness a common trait in the Targaryens. King Jaehaerys II Targaryen once said: "Madness and greatness were two sides of the same coin and every time a new Targaryen was born, the gods would toss the coin in the air and the world would hold its breath to see how it would land".
