An omni-God is a God who has the attributes of omnipotence, omnibenevolence, and omniscience (and perhaps omnipresence).
However, saying these words is not enough to say exactly what they mean.
(Similarly, when a translation of Genesis, say, has God being described as ‘almighty’, what exactly does this mean? It is easy for a contemporary reader, who might be acquainted with the standard classical theological understanding of omnipotence, to ascribe that classical conception to the word ‘almighty’. Yet, that could be an anachronistic understanding – projecting our Aquinas-descended theology onto the Biblical writers, perhaps.)
A classical philosophical understanding of this, or ‘classical omni-God’, can be thought of as involving God not as a being but Being itself, out of which all things share in being. It is not just that they are created at some point in time by God (i.e., whenever He wills it), but also that at any moment their existence is due to God, as God is the base of existence. That is, in this classical conception, there is no being proper except for God’s being. Similarly, God can change all being at any moment. So, this conception of God’s omnipotence involves His being able to do anything ‘in the snap of a finger’, where the only constraints are typically understood as those of basic logic. This conception owes a significant amount to Aquinas and Plato, and is essentially an abstract, philosophically-driven conception of God.
What sort of alternatives are there to this approach? One possibility is what could be called an ’empirical omni-God’. This conception treats highly abstract, philosophical conceptions of God as secondary, and instead seeks to assemble an accurate conception of God’s nature by empirically assembling evidence of various kinds, building up to or thereby inferring certain of God’s attributes (omnipotence of a certain sort, and so on). This view owes a significant amount to William James, and is essentially an empirical conception of God.
So, the classical view is more a top-down conception of omnipotence (and so on), the empirical view is more a bottom-up conception.