Before answering this question let us understand each of them individually. Shiv and Parvati together exist in one form at the Kailash mountain, where Shiv is meditating and Parvati sits next to him.
He is described as:
कर्पूरगौरं करुणावतारं संसारसारं भुजगेन्द्रहारम् ।
सदा वसन्तं हृदयारविन्दे भवं भवानीसहितं नमामि ॥
Meaning: I bow to that Ishwar along with Bhavani (Shiva and Parvati), who is as white as camphor, an incarnation of compassion, the essence of this world, who wears a serpant around his neck and is ever present in the lotus abode of our hearts. Shiv has many incarnations and has mainly taken 10 avatars along with Parvati, 26 leela Avatars alone and is existing in 11 forms as Rudra born of rage and Eight other forms again as Rudra, born crying. He has various other avatars that existed in various kalpas and many avesh and amsha avatars also, for eg. Maharishi Durvasa is an amshakala avatar of Shiv.
Who is Rudra ?
Rudra is a form of Shiv born from the rage of Lord Brahma. He was angry as Narad had distracted Sanat-Kumar from creating srishti as Brahma had instruced them to do so. They chose penance instead of marridge and went to the woods. From Brahma's frown appeared Rudra, Bluish-Red (NeelaLohita)
half-male, half-female and was intructed to seperate. He seperated the female and she is Adi-Shakti. Rudra, further divided himself into 11 forms, some fierce and some calm.
They are called :
From the woman were born the eleven Rudranis who became wives of the Rudras. They are:
Rudra as a kumara and his eight forms. (*Exact translation of Vishnu Puran)
In the beginning of the Kalp, as Brahmá purposed to create a son, who should be like himself, a youth of a purple complexion appeared, crying with a low cry, and running about. Brahmá, when he beheld him thus afflicted, said to him, "Why dost thou weep?" "Give me a name," replied the boy. "Rudra be thy name," rejoined the great father of all creatures: "be composed; desist from tears." But, thus addressed, the boy still wept seven times, and Brahmá therefore gave to him seven other denominations; and to these eight persons regions and wives and posterity belong. The eight manifestations, then, are named Rudra, Bhava, Śarva, Iśána, Paśupati, Bhíma, Ugra, and Mahádeva, which were given to them by their great progenitor. He also assigned to them their respective stations, the sun, water, earth, air, fire, ether, the ministrant Brahman, and the moon; for these
are their several forms. The wives of the sun and the other manifestations, termed Rudra and the rest, were respectively, Suverchalá, Ushá, Vikesí, Sivá, Swáhá, Diśá, Díkshá, and Rohini.
11 Rudra Avatars according to the Vayu Puran
Lord Shiva took birth on the earth as the eleven Rudras from the Kashyap wife’s (Surabhi's) womb. These forms of the eleven Rudras are associated with the battles with demons in the past in order to save the people and Gods. Following are the name of eleven Rudras:
1) Kapali 2) Pingal 3) Bheem 4) Virupaksha 5) Vilohit 6) Shastra 7) Ajapaad
8) Ahirbudhnya 9) Shambhu 10) Chand and 11) Bhav.
Rudra according to grammar translates to "the one who cries." "Rud" the root means to cry and 'ra' is the suffix attached to it in present tense and Karta (Karak-Case).
Who is Shiv ?
Shiv according to Shaiv Philosophy, is the supreme entity or ParamBrahma. According to Vaishnav Philosophy, Shiv is a Tamasi form of Maha-Vishnu, the supreme entity or ParamBrahma. MahaVishnu or AadiNaaraayan, also known as the KaranodakShayiPurush or the Loka-Veda-Prathita-Purush.
अतोऽस्मि लोके वेदे च प्रथितः पुरुषोत्तमः।।15.18।।
Krishna, in the Bhagwad Geeta, proclaims himself as the KaranodakShayiPurush or the Loka-Veda-Prathita-Purushottam. He has 3 parts for creation(Brahma as Rajas form), protection (Vishnu as Sat form) and destruction(Shiva as Tamas form). He himself is responsible for everything good or bad, creation and destruction.
Incarnations and Idols of Shiv
The most important five incarnations are: Tatpurush, Namadeva, Aghoresh, Sadhojat and Ishan.
There are eight idols of the Lord Shiva which includes Ugra, Sharva, Bhava, Rudra, Bheema, Pashupati, Ishan and Mahadeva. Ten fastidious incarnations of the God Shiva known as Dash Avatar and 10 incarnations of the Mata Sakti known as the Dash Mahavidya.Describing concerning to the 10 incarnations and their corresponding power (Shakti), the first incarnation of Lord Shiva was the Mahakal and Shakti was Mahakali. Second incarnation of both of them was Tar and Tara. Third incarnation of both of them was Bhuvaneshwar and Bhuvaneshwari.Fourth incarnation was Shodash and Shodashi or Shri. Fifth incarnation was Bhairav and Bhairavi. The sixth incarnation was Chhinamastak and Chhinamasta. Seventh incarnation was Dhoomvan and Dhoomvati. The eighth incarnation was Baglamukh and Baglamukhi. The ninth incarnation was Matang and Matangi. Tenth incarnation was Kamal and Kamala.
All the leela avatars
Apart from all the above incarnations of the Lord Shiv, he had taken some other (Leela avatars) incarnations which are listed below:
1. Ardhnaarishwar Avatar
2. Nandi Avatar
3. Sharabh Avatar
4. Grihpati Avatar
5. Neel Kanth Avatar:
6. Rishi Durvasha Avatar
7. Mahesh Avatar
8. Hanuman Avatar
9. Vrishabh Avatar
10. Piplaad Avatar
11. Vaishyanath Avatar
13. Yatinath Avatar
14. Krishna Darshan Avatar
15. Awdhuteshwar Avatar
16. Bhichhuwarya Avatar
17. Sureshwar Avatar
18. Bramhchari Avatar
19. Sunatnartak Avatar
20. Saddhu Avatar
21. Ashwathama Avatar
22. Kiraat Avatar
23. Veerbhadra Avatar
24. Bhairav Avatar
25. Allama Prabhu
There are many more avatars that cannot be counted in one page, for eg. Batuk Avatar, to pacify Maata Kaali, Bhootnath Avatar to kill Arunasur, Or Chandrashekhara Avatar when Meenavati fainted looking at Shiva's primitive appereance. I am sorry as they cannot be counted by one person in one life.
Shiva and his 12 Jyotirlingas
Shiv has had 2 Avatars combined with Vishnu. One is Lord Ayyappan or Harihara Avatar and another is Lord Dattatreya as the trinity combined in one body. Shiv in Kaliyug incarnated himself as Adi Shankaracharya.
We should all engage ourselves in reading the scriptures ourselves instead of looking for answers on Tv or the internet. People take advantage of our ignorance and carelessness, thus fooling us and engaging us in materialistic lifestyle. People visit a Guru for getting good marks, or having a good sallery. People should visit a Guru for what you read but did not understand. Do not ask for anything from Bhagwan. He knows what is required and what needs to be done without you telling him.
(Article contributed by Shri Rishi Goswami)