The History Of Worf...

Worf's family ties are slowly revealed across several hundred episodes and the movies. An important aspect to understanding Worf is that he was adopted by Federation parents, so he has both adoptive and biological family.

The House of Mogh was a family of high social and political rank which was for a time represented on the Klingon High Council. In Star Trek VI: The Undiscovered Country (set around 70 years before the Next Generation era began), Colonel Worf (also portrayed by Dorn) appears as the legal advocate of Captain James T. Kirk and Dr. Leonard McCoy after they are accused of killing Chancellor Gorkon of the Klingon High Council. He was also a member of the Klingon delegation at Camp Khitomer. Although not explicitly stated, he was intended to be Worf's grandfather.

Worf has a son named Alexander with a half-human half-Klingon woman named K'Ehleyr, a character introduced in "The Emissary", however she is later killed in "Reunion", a "sequel" to that episode and part of the Worf story arc, leaving Worf as a single parent. 
Alexander later has to live aboard Enterprise-D when K'Ehleyr is killed. After TNG ends, Worf gets moved to the Deep Space Nine space station where he eventually marries the Trill symbiont Jadzia Dax. (see "You Are Cordially Invited...") On DS9 Worf misses the Enterprise-D "family" that he had, often bemoaning the cut-rate work ethic and unfriendliness on the wayward outpost.

The episode "Sins of the Father" introduces Worf's long lost brother Kurn, who is also an orphan of the House of Mogh. His adoptive parents have another son, Nikolai Rozhenko whom Worf grew up with. Nikolai and Worf interact in "Homeward" where it is revealed that Worf will likely have a nephew or niece.

Worf was born in 2340 on Qo'noS as the son of Mogh. Five years later, his parents moved to the Khitomer colony. Worf's parents were killed during a surprise attack by the Romulans on the Khitomer outpost. The colony's distress call was answered by the Federation starship USS Intrepid. Chief Petty Officer Sergey Rozhenko found Worf in the rubble and took him in after failing to find any living relatives. Rozhenko and his wife Helena raised him on a small farm colony on the planet Gault, a world of about 20,000 inhabitants, almost all of them human. Worf also has a human brother, Nikolai, with whom he often quarreled. He also spent time on Earth in his parents' native city of Minsk, later recommending it to Miles O'Brien as one of his favorite places on Earth.

Worf did not take the Rozhenkos' last name, preferring to be addressed by the Klingon designation "Worf, son of Mogh". However, his son Alexander Rozhenko, who was raised by the Rozhenkos after his mother K'Ehleyr died, did use their surname. Although Worf was raised by humans, he considered himself a Klingon at heart and studied the ways of his people. As an adult, his mannerisms and personality, as well as his innate sense of honor, became more Klingon than human.

Worf's brother Kurn, barely a year old at the time of the Khitomer attack, had been left behind on the Klingon homeworld Qo'noS by his parents. Lorgh, a friend to House of Mogh (appearing only in dialog of the episode "Sins of the Father"), was charged with the care of the younger son, originally expecting Mogh's stay at the Khitomer outpost to be short-term. Lorgh adopted Kurn after the attack, but informed Klingon authorities that he had died with the rest of the family. Kurn was not revealed as being alive until both brothers were adults.

In 2357, Worf entered Starfleet Academy. He graduated in 2361 and was commissioned with the rank of Ensign, becoming the first Klingon officer in Starfleet. Although Worf took immense pride and a sense of honor from serving in Starfleet, most other Klingons shunned and belittled his choice of vocation.