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 :

Manyu
Manu
Mahmasa
Mahan
Shiva
Ritudhvaja
Ugraretas
Bhava
Kama
Vamadeva
Dhritavrata

From the woman were born the eleven Rudranis who became wives of the Rudras. They are:

Dhi
Vritti
Ushana
Urna
Niyuta
Sarpish
Ila
Ambika
Iravati
Sudha
Diksha

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.

"यस्मात्क्षरमतीतोऽहमक्षरादपि चोत्तमः।
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

26. Khandoba

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
1. Somnath
2. Mallikarjuna
3. Mahakaleshwar
4. Omkareshwar
5. Vaidyanath
6. Bhimshankar
7. Rameshwar
8. Nageshwar
9. Vishwanath
10. Tryambakeshwar
11. Kedarnath
12. Ghushmeshwar

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)

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