Krishna's place in Hindu theology turns on the idea of avatara, the descent of the divine into the world, and on a long argument about just how much of the divine he is.
The doctrinal charter is a couplet in the Bhagavad Gita (4:7-8): whenever righteousness declines and unrighteousness rises, the Lord says, he comes into being, age after age, to protect the good and restore dharma. On the standard Vaishnava scheme Krishna is one of the ten avatars (dashavatara) of Vishnu, alongside Rama and the rest. In this reading he is a supremely important form of a greater deity.
But an influential strand goes further. Gaudiya Vaishnavism, the tradition of Chaitanya in sixteenth-century Bengal, reads a line of the Bhagavata Purana, krishnas tu bhagavan svayam, "but Krishna is the Lord himself", to mean that Krishna is not an avatar of Vishnu at all but the original, complete Godhead (svayam bhagavan), of whom Vishnu is a partial expansion. The difference is not pedantic: it decides which name sits at the top of reality, and comparable claims are made for Krishna in other Vaishnava lineages.
This is why Krishna can appear in the tradition both as a figure within a pantheon and as its absolute centre, depending on who is speaking. It is the same doubleness that lets the great commentators read the Gita's "Lord" toward opposed conclusions about the relation between God, the self and the world, a divinity defined as much by descent into the world as by transcendence of it.