Arca-vigraha – also murti; Deity of the Lord (in the temple) through which the Lord reveals Himself to His devotees and enables them to serve Him personally.
Since we are not (yet) able to perceive the pure spiritual form of the Supreme Personality of Godhead, He has manifested His presence through material elements such as a painting or a murti.
So we can see your beauty and meditate on it. In this form the Lord personally accepts the worship of his devotees.
In the Vedic understanding there is a difference between (monotheistic) worship of God, worship of gods and worship of idols.
The Sanskrit word for deity is “deva“. There are many deities in the Vedas such as Brahma (the architect of the universe), Indra (the king of a heavenly planetary sphere who also has great influence over the weather), Surya (the deity of the sun), etc. However, Krishna is mentioned in the Vedas as referred to solely as "devadeva", the deity of all other deities, or the god of gods. This One Supreme God is the origin of all other powerful beings. Some also speak of God and the demigods to further emphasize the difference.
The moment I worship the demigods, while being aware that there is also the One Supreme God who is above all others, it is god worship (some say demigod worship).
On the other hand, the moment I mistake the deva (i.e. demigod) for the One Supreme God, it is idol worship - for idol worship is misregarding and worshiping a being that is not God as God.
Even the Bible does NOT say that there are no other gods besides Yahweh. These gods exist. One should only understand which god is on which level. The Ten Commandments say you shall worship no other gods besides Me. So nothing is said about these other gods not existing. So we are also dealing with a monotheism there, which does not deny, however, that there are also other demigods and powerful beings. The Ten Commandments only command observance of proper hierarchy.
(from gour-ni-times.de, by Paramasrehya dasa)
The form of God is beyond human imagination, but when God reveals himself and the scriptures give descriptions of how the form of God is to be worshiped, it is a different matter.
Srila Prabhupada describes the authorized form of Deity worship using the mailbox analogy. This can be explained as follows:
The post office may be far from my home, so the post office sets up various mailboxes in the neighborhood. If I now paint a box yellow myself and write mail on it, then this fake mailbox has no value. The post box set up by the post office, on the other hand, is fully authorized and in a sense not different from the post office. Similarly, the Deities of God are not simply made according to the artist's imagination, but according to the instructions of the scriptures, which contain very precise instructions as to proper posture, hand positions, bodily proportions, and so on. After this there is an extensive dedication ceremony in which the Lord is asked to imbue the Deity with His divine presence.
From that point on, God is worshiped in a very comprehensive way in the form of the Deity. The Deity is a divine incarnation with which the Lord entrusts Himself to human care. The image is therefore a divine guest and should be treated as such. Therefore incense, flowers, lights, song and food are offered to Him. Through this service, a loving exchange can develop between God and his devotee.
(Krishna and his siblings - appeared in this world around 5000 years ago)
Three Deities of God as worshiped in the coastal city of Puri in Odhisa, India for millennia. They depict Krishna together with his brother and sister, who, with their wide eyes and broad laughter, express ecstatic feelings in the separation of their devotees.
Jagannatha means the lord (natha) of the universe (jagat) and refers to the Supreme Personality of Godhead, Sri Krishna.
Baladeva, the god (deva) of power (bala) or Balaram, is Krishna's elder brother and his first immediate extension.
Subhadra, the auspicious, is Krishna's sister, embodies the mystical power of God (yoga-maya) and is the wife of the great Pandava warrior Arjuna.
(Krishna & Balaram – appeared around 500 years ago)
Sri Sri Gaura Nitai are considered channa-avatars, secret, hidden incarnations of God.
Sri Krishna appeared as Sri Caitanya, the son of Mother Saci and Balarama appeared as Nityananda.
Sri Caitanya Mahaprabhu, Caitanya=Consciousness, (1486-1533) is a unified embodiment of Radha and Krishna. He is considered a hidden incarnation because he does not openly reveal his identity as the incarnation of God, but plays the role of a devotee (bhakta avatar). The scriptures describe Caitanya as the 'golden avatar of Kaliyugas' who, through the chanting of the Holy Names of Krishna, brings pure love of God into the world.
Sri Nityananda, eternally (nitya) immersed in bliss (ananda), is the transcendental brother of Caitanya, who is regarded as a more merciful incarnation of God and who blesses everyone who chants the holy name of God with confidence.