This is Siva in his aspect as the universal teacher, teaching the secrets of yoga, tantras, yantras, alchemy, magic, occult knowledge, arts and sciences, ancient history or knowledge of the future to the sages and saints, gods and goddesses and his highly qualified devotees.
He is called Dakshinamurthy, because he does his teachings sitting on the snowy mountains of Himalayas and facing towards the Indian subcontinent, which is in the southerly direction to him.
The images of Dakshinamurthy, depict Siva in his pleasant mood, seated on a high seat, with one leg folded while the other rests on the Apasmarapurusha, the deluded self.
Two of his arms hold a snake or rosary or both in one hand and fire in the other. The snake is a symbol of tantric knowledge and the fire is a symbol of enlightenment. Of the remaining two one is in abhayamudra (posture of assurance) and the other holds a scripture in gnanamudra (posture of presenting knowledge).
This image signifies the importance of Siva in the form of Linga as the Supreme Self, without a beginning and without an end. According to Hindu mythology, Siva once revealed his infinity to Brahma and Vishnu in the form of a pillar of fire that could not be scaled by either of them from one end to the other. As Lingodhbava-murthy, Siva appears seated in the heart of a Linga, with four arms, while Brahma and Vishnu adore him from the two sides.
This is Siva in his ascetic aspect, wandering from place to place, with a begging bowl made of human skull, doing penance or lost in his own thoughts. Even today we can see some followers of Siva going around the villages in India in this form. Some of them even do a little magic to attract our attention or scare away the trailing children.
This is Siva in a mood of reconciliation and friendship with Vishnu. Also known as Harihara or Sankaranarayana. The images show the right half of Shiva on the right side of the image and the left half of Vishnu on the left side.
This Siva and Parvathi together in one form signifying the unity of Purusha and Prorate. The feminine left half of Parvathi is fused with the masculine right half of Siva in one continuous form, sometimes standing with the Bull Nandi in the background, or sitting on a pedestal and blessing the worlds, with eyes open or closed.