Process Theology means different things to different people. Generally speaking, it refers to the idea that God is both wholly other yet immanent within a process of creation.
Christian versions emphasize a God who is, on the one hand, eternal, unchanging and beyond, and yet who also feels and is affected by the actions of humanity.¹
According to this view, God suffers with humanity, leading individuals to eternal salvation not through coercion but as a loving parent, friend or spouse.
Wikipedia says that process theology comes out of the work of the philosopher Alfred North Whitehead, but this claim could be a bit misleading if taken out of context. Reading further down the Wikipedia entry, we see that Whitehead is influenced by a whole host of Medieval theologians and ancient philosophers—St. Thomas Aquinas, St. Anselm, Aristotle, to name just a few.
The same entry at Wikipedia sums up the general Christian view of process theology very nicely:
Rather than see God as one who unilaterally coerces other beings, judges and punishes them, and is completely unaffected by the joys and sorrows of others, process theologians see God as the one who persuades the universe to love and peace, is supremely affected by even the tiniest of joys and the smallest of sorrows, and is able to love all beings despite the most heinous acts they may commit. God is, as Whitehead says, “the fellow sufferer who understands.²
¹ Those interested should look at the discussion of Dipolar Theism.
² See this entry for some of the variations present in Christian and non-Christian faith groups: https://en.wikipedia.org/wiki/Process_theology
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