The most famous sons of Shiva are Ganesha, Karthikeya and Ayyappa. He has other sons and daughters, who are lesser known. We will look at them one by one.
Ayyappa /Sastav, who is now enshrined in the famous temple of Sabarimala in Kerala is an offspring of Shiva and Vishnu, when he took the form of beautiful Mohini to lure the asuras away from Amrut, an elixir which was the output of the churning of the ocean of milk and also to kill Narakasura. Ayyappa was a great warrior trained by Parashurama.
Andhaka, as the name suggests was blind, was a son of Shiva. He was raised by a demon named Hiranyaksha who was childless. He was killed by Shiva himself when he tried to molest Parvathi.
Bhauma was another son of Shiva. He was born when three drops of Shiva's sweat fell on earth /Bhumi. He was a beautiful red-skinned infant with four arms. Since Shiva was in deep meditation Bhumi raised the kid. When he grew up he did severe penance to please Shiva and Shiva granted him the Mangala Loka. He established himself in the Solar System as planet Mars/ Mangala.
Khuja is also Shiva's son, who is worshiped as the 'Lord of Iron'. Bhauma and Khuja appears to be one and the same.
Nandi was also Shiva's son. In Shiva Purana, as a boon Shiva promised Shiladra that he will be born as his son.
Veerabhadra was born of a lock of Shiva's hair. He destroyed Daksha's yagna.
Ganesha, Shiva's most popular son. The elephant headed God.The most intelligent /illuminated one. The remover of sorrows and hurdles. Ganesha wrote down the epic Mahabharatha, while Vyasa dictated it. Padmapurana says Ganesha was married to sisters Riddhi and Siddhi ( Siddhi and Buddhi ). It is said that you should not keep more than one Ganesh idol in your Puja Room.
Karthikeya was born to slay the demon called Tarak and his army. He is worshiped more in South India where he is known as Murugan /Subhramanian /Senthil etc. His appearance is that of a teenager holding a spear, with a peacock and a snake on the ground -symbolizing the two extremes of spirituality- Shiva and Shakti. He was the incarnation of an already illuminated soul. He is also called Guru.
Ashoka Sundhari:- When Lord Ganesh and Karthikeya were grown up, maa Parvathi started feeling lonely. She went to the famous Kalpavriksha tree and asked for a boon and wished for a daughter. Thus Ashoka Sundhari was born. Ashoka meaning, she removed the 'shok' or sadness from Parvathi's life and Sundhari because she was a beautiful young girl. Ashoka Sundhari was later married to Nahusha of the lunar dynasty. Ashoka Sundhari is considered to be the mother of Yayati and a hundred daughters. Devi Ashoka Sundhari is worshiped in Gujarat. She is associated with salt.
Jyothi is the goddess of light.The deity can be found in a few temples in Tamil Nadu. She emerged from Shiva's halo and is a physical manifestation of His grace.
Mansa /Manasa is a goddess who cures snake bites in Bengali folk lore. She is the sister of Vasuki, the king of snakes. She was born when Shiva's semen touched a statue carved by Kadru, the mother of snakes. She was a very unhappy and bad tempered goddess. Parvathi /Chandi, her stepmother hated her. Chandi burnt one of her eyes, so she is also known as the 'one-eyed goddess'.She is also the goddess of fertility and marriage rites. She was the one who saved Shiva when he drank poison. She is depicted as a woman covered with snakes sitting on a lotus / standing on a snake under a canopy of seven cobras. Sometimes she is depicted with a child on her lap, believed to be her son.