"Mundungus Fletcher’s put in a claim for a twelve-bedroomed tent with en-suite Jacuzzi, but I’ve got his number. I know for a fact he was sleeping under a cloak propped on sticks."
-- Percy Weasley, after the Quidditch World Cup (GF10)
Mundungus Fletcher is a rather dodgy character, a crook whose shady business dealings are always getting him into trouble. However, he’s also a member of the Order of the Phoenix.
Harry and the Weasley boys think he is a lot of fun to spend time with, since he can tell stories that will keep his listeners in stitches. He drinks quite a bit and swears magnificently. Albus Dumbledore seemed to have the attitude that even though he's a crook, he can be mighty handy to have around sometimes since he hears things others don't (OP5).
Mundungus, or "Dung" as he's called, smokes a grimy, smelly pipe. Molly Weasley doesn't like him smoking the pipe in the kitchen (OP5). She doesn't like him at all, come to that.
Dung and the Hog's Head
In the 1970s Mundungus was thrown out of the Hog's Head Pub, so when he was assigned to keep an eye on Harry 20 years later, he was forced to hang around that pub dressed as a witch (OP16). Fortunately, keeping one's face hidden is a bit of a fashion in the Hog's Head. Dung may have other Hog's Head connections: the day Harry caught Dung with peddling Black family heirlooms, he was seen speaking withAberforth, the Hog's Head barman (HBP12). (HBP12).
At the Quidditch World Cup: merely fraud and an attack on Arthur Weasley
At the Quidditch World Cup, Fletcher was sleeping under a cloak propped up with sticks. After the Death Eater incident, however, he tried to put in to the Ministry for compensation for a 12-room tent with en suite jacuzzi (GF10). During a Ministry raid (presumably searching for misused Muggle artifacts), Mundungus tried to hex Arthur Weasely when his back was turned (CS3).
Guarding Harry or "business opportunity?"
On August 2, 1995, Mundungus Fletcher was on guard duty, tailing Harry Potter around the neighborhood of Privet Drive. He got wind of a business deal on some stolen cauldrons, however, and he abandonded his post. During his absence, Harry and Dudley were attacked by two Dementors (OP1-2). Dumbledore was furious with him and Arabella Figg beat him around the head and body with her string shopping bag full of tins of cat food. He was only somewhat abashed, however, and later, when he met Harry, he tried to explain along with his half-hearted apology (OP).
Purple robes and Tentacula seeds
Needless to say, Molly doesn't care for Mundungus at all, especially after he abandoned his post guarding Harry during the summer. However, during the cleaning of 12 Grimmauld Place, Dung rescued Ron from some particularly nasty purple robes, and Molly was somewhat mollified (OP6). It is doubtful, however, that Molly would have approved when Mundungus sold illegal Venomous Tentacula seed pods to Fred and George (OP).
Sent to Azkaban
In the early spring of 1997, Dung is arrested and sent to Azkaban for "impersonating an Inferius during an attempted burglary." (HBP21).
During the Battle of Seven Potters, Dung was Polyjuiced into Harry and rode a broomstick behind Mad-Eye Moody. When Voldemort attacked, he Disapparated, leaving Mad-Eye Moody to be killed (DH4).
Fencing stolen goods, eavesdropping, fraud, swearing, burglary, telling a good story (probably embellished).
Other canon notes and references
Ancestry: half-blood (HBP13).
Skills: Fencing stolen goods, eavesdropping, fraud, swearing, burglary.
Location of home: Unknown.
Alliances: Order of the Phoenix (OP2-8), and who knows what else.
Pipe: Emits greenish smoke that smells like burning socks (OP5).
Hair: Straggly ginger hair (OP2).
Eyes: Bloodshot and baggy (OP2).
Other characteristics: Tattered clothing, unshaven, smells of stale alcohol and tobacco. Short; shorter than Harry at 17 (OP2, DH4).
"Mundungus" according to JKR, is the name of a particularly smelly variety of tobacco (RAH).
"Fletcher" = [English] one who makes arrows (the feathers are referred to as fletching).