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Lingodbhava
Lingodbhava murthi is an iconic representation of Shiva, installed in the rear niche (devakoshta) of the sanctum (garbagriha) enshrining a Shiva Lingam. Since most Temples face east, Lingodhava faces West.
The story of Lingodbhava is that of the attempts of Vishnu and Brahma to discover the origins of Shiva as a column of fire. Brahma assumed the form of a swan and flew upwards, while Vishnu assumed the form of a boar, and burrowed down into the earth. Days of search in either direction proved futile, and hence the duo surrendered to Shiva.
The non-anthropomorphic form Shiva Lingam is a representation of this infinite cosmic column of fire, whose origins were not traceable by Brahma or Vishnu. The Shiva lingam is the centre of reverence and worship in all Saivite temples. This legend has particular resonance at Arunachala, as it is held that Arunachala Hill itself, is a manifestation of the cosmic column of fire. (Tiruvannamalai, is one of the Pancha Bhuta Stalas, representing the primordial element fire).
There is another interesting aspect to this legend. While Brahma was flying upwards in the guise of a swan, he saw the petals of a ketaki flower drifting down. Tired by the futility of his efforts to reach the top of the mysterious column of fire, Brahma requested the flower to acquiesce to his lie that he had seen the top of the column where the flower had previously resided. Accompanied by his accomplice, Brahma confronted Vishnu and asserted that he had indeed discovered the origin of the cosmic column. An enraged Shiva appeared out of the fiery column and cursed Brahma so that he would not be worshipped in Temples on earth. Thus there are no Temples dedicated to this God of any significance in India.
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Bhikshatana
Bhikshatana is a form of Lord Shiva, when he manifested as an ‘enchanting mendicant,’ a wandering ascetic. The Rishis in Darukavana had become convinced that action was supreme and that there was no need to worship God. To teach them the value of worshipping God, Shiva manifested as a naked mendicant, begging for alms. The wives of the Rishis were enchanted with the naked beggar and followed him. The Rishis quarrelled with Shiva but the matter was resolved when the Rishis were made aware that they were fighting with the Supreme God. As Bhikshatana, the Lord is depicted as naked, with a snake around the hips and sandals on his feet.
Check this link to read about ‘Mohini and the Sages’ (under Thiruvoodal Sagas) as one of the causes for a tiff (i.e. tiruvoodal) between Lord Shiva and Parvati.
Still another version of the legend is that Lord Shiva had to wander as a mendicant in order to expiate the sin of having severed the head of Brahma, till he reached the Himalayas, where he was relieved of his sin.
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Shiva as Ardhanarishvara
In Hinduism, Ardhanari or Ardhanarishvara, is an androgynous deity composed of Shiva and his consort Shakti, representing the synthesis of masculine and feminine energies. The Ardhanari form also illustrates how the female principle of God, Shakti is inseparable from the male principle of God, Shiva. Ardhanari in iconography is depicted as half-male and half-female, split down the middle.
The term 'Ardhanarishvara' is a combination of three words- 'ardha', 'nari' and 'ishvara', meaning respectively, 'half', 'woman' and 'Lord' or 'God', that is, Ardhanarishvara is the Lord whose half is woman, or who is half woman.
Go to this link to read about Arunachala and the legend of Ardhanarishvara and this link to read about Deepam Ardhanarishvara.
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Dakshinamurthi
Is an aspect of Shiva as a guru (teacher) of all type of knowledge, particularly jnana. This aspect of Shiva is his personification as the supreme or the ultimate awareness, understanding and knowledge and also represents Shiva in his aspect as a teacher of yoga, music, wisdom and giving exposition on the shastras.
Dakshinamurti literally means 'one who is facing south'. South is the direction of Death, hence change. In every Siva temple the stone image of Dakshinamurthi is installed, facing south, on the southern circumambulatory path around the sanctum sanctorum. Perhaps, of all Hindu Gods, he is the only one sitting facing south.
In his aspect as Jnana Dakshinamurti, Shiva is seated upon a deer-throne and surrounded by sages who are receiving his instruction. He is shown as seated with his right foot on a demon (which represents ignorance) and his left foot lies folded on his lap. In his upper arms, he holds a snake or rosary or both in one hand and a flame in the other; while in his lower right hand is shown in vyakhyanamudra, his lower left hand holds a bundle of kusha grass or the scriptures. Dakshinamurthi is portrayed as being in the yogic state of abstract meditation - and as a powerful form brimming with ever flowing bliss and supreme joy.
Significance
Go to this link to read some words by Sri Ramana Maharshi about Dakshinamurti.