The Name
The Lord has numerous names, according to the pastimes He engages in or due to some of the qualities He possesses. The name ‘Krishna’ means ‘all-attractive’. It is the most complete and appropriate name that refers to God because it gives a true description of God.

God is said to possess all opulences in full, the six main attributes being; strength, fame, knowledge, beauty, renunciation and wealth. Human beings might also possess some of these attributes, but only in minute proportions. However, Krishna has all of them and that also in infinite proportions. Therefore, He is named ‘Krishna’- the all-attractive.

His position
Why, among all the ‘Gods’ and ‘Goddesses’ found in the Hindu pantheon, do we accept Krishna as God?

ete camsa-kala pumsa krsnas tu bhagavan swayam
Indrari-vyakulam lokam mrdanyanti yuge yuge

All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of te Lord, but Lord Krishna is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incanrates to protect the theists. Srima-Bhagavatam 1.3.28.

In the Srimad-Bhagavam, it is described in clear terms, that God has many expansions and incarnations, but among all of them, Krishna is the original Supreme Personality of Godhead.
Moreover, working under Him, we find a hierarchy of demi-gods, each of them in charge of a particular aspect of material nature. In the Bhagavad-gita, the demi-gods (or devatas) are referred to as the ‘administrators of material affairs’.

Krishna oversees material nature and activities, but does not take part in them personally. These services have been delegated to these empowered administrators.

His pastimes

Yada-yada hi dharmasya glanir bhavati bharata
abhyutthanam adharmasya tadatmanam srjamy aham

Whenever and wherever there is a decline in religious practice, O descendant of Bharat, and a predominant rise in irreligion - at that time I descend Myself.(Bhagavad-gita 4.7)

The foremost consideration of the Lord is to take care of His devotees. Therefore, He either descends unto the earth or expands Himself to participate in the pastimes.
When Lord Krishna descended unto the earthly planet, during the Dvapara-yuga to enact some pastimes for the pleasure of His devotees, He came in His original, spiritual form. He never took up a material body. He also left the planet out of His own sweet will.

janma-karma ca me divyam evam yo vetti tattvatah
tyaktva deham puar jamma naiti mam eti so Arujna

One who know the transcendental nature of My appearance and activities does not, upon leaving the body, take his brith agian in this material world, but attains My eternal abode, O Afjuna. (Bhagavad-gita 4.9)

His body was spiritual, not material like ours. It was not affected by old-age, disease or death. Even His birth was transcendental, because He first appeared in His fully-grown, Vishnu form in front of His parents, before assuming the form of a baby. During His stay on earth (over a 100 years), He maintained the body of a 16- year old youth. During His stay on earth, He enacted many wonderful pastimes; the glories of which are still being relished by several generations of devotees.