The Lord of the worlds, of the deva-s, and of dharma.
Om lokAdhyakshAya namah.
adhyaksha means master or superintendent. Loka refers to people whose duty it is to practice dharma, sura-s refers to devas who are worshipped with the dharma, and dharma signifies the means by which this worship is performed. BhagavAn in the form of aniruddha is the presiding deity over all these, who closely observes all these and grants the fruits of the dharmic acts. He also ensures that the benefits are not obtained by those who do not observe the dharmic acts, and ensures that those who observe the dharmic acts do not go without the fruit.
An alternate interpretation is that loka refers to the people, sura-s are those who protect the people, and dharma is the means by which they protect them. BhagavAn is the controller and supervisor or master of all these. For instance, the deva-s control the natural elements and ensure that these perform in such a way that they benefit the people. Thus the rising of the sun, the rains, etc. are controlled. When the deva-s swerve from their path, BhagavAn takes over and ensures that the violations are set right. For instance, when indra got angry with the gokula-vAsi-s and tried to destroy gokulam with unending rain, Lord KRshNa bore the govardhana mountain and protected the people from indra's fury and brought indra under control.
a) The grantor of fruits that are this this-worldly as well as those that are eternal.
SrI Sankara interprets kRta as effect (kArya rUpa) and akRta as cause (kAraNa rUpa), and thus he interprets this nAma as symbolizing that bhagavAn is both cause and effect of all things. KRtam is vyaktam (that is manifested), and akRtam is unmanifested or avyaktam. Or, the nAma can mean that bhagavAn appears with a form that is akRtam or nityam in nature, and also in forms which remain only for a limited time (kRtam).
A third interpretation is that He gives benefits which are this-worldly (kRtam), as well as eternal (akRtam), and so He is kRtAkRta.
One whose Self has a four-fold manifestation.
Om caturAtmane namah.
One explanation for this is the Self has control over the four-fold nature - the wakeful state, the state of dreams, dreamless deep sleep, and turIya state where even the breath is suspended. These involve the external senses, the mind, breath, and suspended breath. These are also the four stages of development which a worshipper who meditates upon the Lord goes through.
SrI Sankara interprets the nAma as signifying that bhagavAn expresses His energies in four forms in each of His acts of creation, protection, and dissolution. He quotes the following sloka in vishNu purANa -
brahmA dakshAdayah kAlas_tathaivAkhila jantavah | vibhUtayo hareretA jagatah sRshti_hetavah || vishNur_manvAdayah kAlah sarva_bhUtAni ca dvija | stither_nimitta_bhUtasya vishNoretA vibhUtayah || rudrah kAlotankAdyASca samastAScaiva jantavah | caturdhA pralayAyaitA janArdana-vibhUtayah ||
The four energies of Hari for creation are - brahmA, daksha and others, time, and all creatures. The four energies of Hari for protection are - vishNu, manu and others, time, and all creatures. The four energies of JanArdana for dissolution are - rudra, time, the god of death, and all creatures. Thus He is caturAtma at all times.
One with four forms (the vyUha forms).
Om catur-vyUhAya namah.
This nAma is interpreted as signifying the four vyUha emanations (vAsudeva, samkarshaNa, pradyumana, and aniruddha). SrI Sankara give the following reference to support this interpretation. In my version of the vyAkhyAna, the original source for this reference is not given, but the quote is attributed to vyAsa -
vyUhyAtmAnam caturdhA vai vAsudevAdi_mUrtibhih |
vishNu, of excellent fame, manifests Himself in four forms, vAsudeva, etc. (vAsudeva, samkarshaNa, pradyumna, aniruddha) and carried on creation etc.
vyUha refers to a form that has a purpose associated with it. pradyumna is for "padaippu" or creation; samkarshaNa is for samhAra or destruction; aniruddha is for protection; vAsudeva is the overall leader of these three forms. Pradyumna, the one in charge of "padaippu" or creation, is endowed with the leadership of aisvarya and vIrya. pra-dymna means one who has enormous vIrya in His responsibility of creation. aniruddha, in charge of protection, is endowed with immeasurable Sakti and tejas. a-niruddha means One who cannot be obstructed in His function of protection. SamkarshaNa is endowed with enormous ~jnAnam and balam. Sam-karshaNa means one who attracts (note AkarshaNa - attraction) or draws everything towards Him and makes them draw into Him during the pralaya or great destruction. VAsudeva is the form endowed with all the six qualities and is the Supreme parabrahman.
He of four teeth.
Om catur-damshtrAya namah.
damshtra refers to canine teeth. The reference here could be to the four full and strongly developed, powerful, beautiful canine teeth in His incarnation in nRsimhAvatAra. SrI Bhattar points out that catur-damshtratvam is a mahApurusha lakshNam. He gives reference to the description of Sri Rama by Hanuman to SItA devi - catur-daSa sam-dvandvah catur-damshtrah catur-gatih - One who has the fourteen parts of the body such as the legs, the hands, etc. fully symmetric with no blemishes, One who has four beautiful canine teeth, and One who has the beauty of the walks of elephant, tiger, lion, and bull. (VAlmIki RamayaNam Sundara kANdam - 35.19).
One with four arms.
Om catur-bhujAya namah.
The four hands carry the conch, the discus, the mace, and the lotus respectively. SrI ChinmayAnanda gives a very nice explanation of how the four arms are used by the Lord in maintaining dharma. The conch calls man to the righteous path that directly leads to peace and perfection, the divine vishNu padam. Not very many of us listen to this inner voice of conscience, the sound of the pAncajanya conch, and so He gently wields the 'Mace' and we come to suffer small calamities and tragic jerks in our smooth existence. If the individual still does not listen to the call of the 'Conch', the 'Chakra' - the wheel of time, annihilates the entire being. This call and punishment are solely meant to take man to the Ultimate Goal, symbolized by the 'Lotus' in His hand.
SrI RAdhAkRshNa SAStri gives the explanation that the four hands signify that He gives the four purushArta-s (I assume dharma, artha, kAma, and mokha are the four being referred to here). Or the four arms represent the four tatva-s - satva, rajas, tama, and ahamkAra. "satvam rajas tama iti aha~nkAraScaturbhujah (gopAlottaratAmini 55).
Yet another explanation is that the Self functions in a four-fold pattern involving mind, intellect, cit, and ego, and these are represented by the four arms containing the conch, the mace, the cakra, and the padma. The cit is represented by the Lotus or Padma, the intellect is the 'Conch', ego is the 'Mace', and mind is the 'Discus'.