AppearanceThe big, red nose on his face is, in fact, his real nose; this is a fact he is very touchy about, going so far as to destroy an entire town when a couple of kids made fun of his nose. Occasionally, he will mishear what others say as an insult about him or his nose, possibly due to paranoia.
In his first appearance he had crossbones going down his face forming an X; he also had two blue lines near both of his eyes, red lipstick and his jolly roger (with a clown nose) on his hat. However, in later episodes after the Loguetown arc his look changes. First a green star in the middle of the forehead; two half circles beneath the eyes, and a straight line on the middle of his chin. Now has two green dots on both sides of his head and four big green dots on the rim of his hat, which still has the jolly roger on it. He also has green eyeliner and green lipstick, with a curvy line going down both sides of his face.
Explanation of Buggy's ponytails
It was revealed in an SBS question that the blue tassels coming from the sides of Buggy's hat is his actual hair, in the shape of two ponytails. While in the confines of Impel Down, Buggy's long hair is tied in just one ponytail.
PersonalityBuggy will often feign intelligence, only to have his true incompetence revealed. Although he is not "stupid" in the traditional sense, he is prone to gross misjudgments that cost him; usually this is due to his overconfidence. In the original Japanese version of One Piece, he inserts hade ni (派手に, "flashy"), into almost every sentence he speaks ("Die with flash! You flashy bastard!").
He has a number of running gags in the series, including his habit of not noticing who he is speaking to until moments after greeting him (for example, when Ace appeared on his ship, Buggy first greets him politely. Then, when he realizes he is on his ship, he blurts out "Who the hell are you?"). He also has what some might consider "dumb luck"; no matter what happens to him he is lucky enough to survive somehow (including being struck by lightning at Loguetown, or being chased by Smoker).
Buggy is one of the few villains to become reformed after facing Luffy, though only slightly. He also begins to accept people's kindness, though he can still be cruel to anyone who isn't a member of his crew or is aiding him in his goals. He also is fun loving and displays a huge fondness for parties, admiring those who are equally fun loving (strangely, he shares this trait with Shanks). He takes great joy in riding on the beliefs of others and took a great advantage of the cheering crowds of prisoners from Impel Down upon his escape, who were impressed with his history on Roger's crew, as well as his connections with Shanks.
He also has a strong love of treasure. Unfortunately (for him) his obsession with treasure led to him leaving himself open in his first encounter with Luffy to attack Nami who was stealing them and was arguably the sole reason for his loss. Buggy fails to understand how anything but gold, silver or jewels can be considered "treasure"; an opinion which apparently hasn't changed in over 22 years even though so far it has led to constant failure.
His primary goal right now is revenge against Luffy, for defeating him, and Shanks, for ruining his dreams during his youth. His dream is to become Pirate King and own all the treasures in the world. After the escape from Impel Down, with the encouragement of his followers, he now has his sights on defeating Whitebeard and conquering the "top".
CrewOf the Jolly Roger Pirates, most relationships between Buggy and the other crewmates remain unknown. Though it has been shown through flashbacks that he would fight with Shanks over just about anything. The Marines themselves describe their relationship as a 'brotherly bond' and Whitebeard himself had noted the pair were always together. Though he was often punished by Rayleigh, he still has much respect for the former first mate of the crew, Silvers Rayleigh. 
Before his loss to Luffy he was willing to use his own crew as human shields, while afterwards he shows a lot more care about their well being.
FriendsHe was good friends with Ace. He also forms a bond with Mr. 3 during their time in Impel Down. Due to the fact that he helped free the prisoners from Impel Down (though he was only planning to use them as sacrifice for his own escape) many of them have earned a deep respect. Since finding out about his Legendary connections, most of the prisoners have seen to see him as a Messiah, and are willing to follow him anywhere, a fact which he uses to his advantage. He is also good friends with Gaimon. However they battled first when he crashed on the island. Neither of them knew the others relationship with Luffy.
EnemiesWhile he hates Luffy, he has proven that though willing to exploit Luffy's naive ways he will work with him to his own goals if he is to benefit from it. He is also not above exploiting his previous "friendship" with Shanks whom he discredits now the pair have gone their separate ways. Interestingly, Buggy's current relationship with Luffy is similar to his past relationship with Shanks: in a slightly friendly antagonistic way.
Abilities and PowersWhile not as powerful as other former Jolly Roger pirates. Buggy is a fairly capable fighter who was strong enough to survive all the battles with Whitebeard's crew and the extreme dangers of the Grand Line but his arrogance and greed usually gets the best of him, preventing him from fighting at his fullest.
Devil FruitBuggy accidentally consumed the Bara Bara no Mi, a Devil Fruit which allows him to separate his body parts and control them independent of each other, but robbed him of the ability to swim, which, according to Shanks (back when the two were still cabin boys at Roger's ship) he was excellent at. The abilities gained from this fruit also make Buggy immune to slashing, cutting and cleaving attack; any such attack just separates him into his component parts.
In addition, most of his body is capable of flying through the air, with enough force to send a man flying through a wall - with the crucial exception of his feet. Even though the parts are separate, Buggy can still feel what the part feels; e.g. when Luffy attacked Buggy, he separated his torso from his waist, only to find that Luffy was aiming for the crotch (needless to say, this left him with a rather painful result). Also, when Buggy used Bara Bara Festival in an attempt to take down Nami, Luffy took one of the feet and tickled it, causing Buggy to break down in laughter. No matter how much of his body he separates, something also seems to act normal; e.g., his feet walking as normal while Buggy sends the rest of his body flying. Though Buggy is a strong fighter and his Devil Fruit power makes him almost invincible, he was constantly leaving himself open in the latter half of his match for Luffy due to distractions, resulting in many careless mistakes.A weakness of his fruit is that if a component part is restrained, he can't use the component part, shown by Nami when she restrained a part of his body in their first fight with him.
The major flaw of his devil fruit's ability is his feet. Should his feet not remain on the ground firmly, he cannot separate the components of his body. Therefore, he cannot split his body whilst airborne.