"Deep time" refers to the time scale of geologic events, which is vastly, almost unimaginably greater than the time scale of human lives and human plans. It is one of geology's great gifts to the world's set of important ideas.
Many ancient traditions asserted that the universe is not only much larger than what we see, but much older too. The Hindu series of yugas, for example, employs lengths of time so great as to be meaningless in human terms, a way of suggesting eternity through the awe of large numbers. But the Judeo-Christian Bible described the history of the universe as a series of specific human lives, starting with "Adam begat Cain," between the creation and today. Bishop James Ussher, of Trinity College in Dublin, made the definitive version of this chronology in 1650 and announced that the universe was created starting in the evening of 22 October in 4004 BCE. Whether you regard that as a dangerous presumption upon awe or a helpful guide to cosmology, the biblical chronology was good enough for people who had no need to measure geologic time.
The Scottish geologist James Hutton is credited with exploding that young-Earth chronology with his painstaking observations of his farm fields and, by extension, the surrounding countryside. He watched the soil being washed into local streams and carried out to sea, and imagined it slowly accumulating into rocks like those he saw in his hillsides. He further supposed that the sea must exchange places with the land, in a cycle designed by God to replenish the soil, so that the sedimentary rock on the ocean floor could be tilted and washed away by another cycle of erosion. It was obvious to him that such a process, taking place at the rate he saw in operation, would take an immeasurable amount of time. Others before him had argued for an Earth older than the Bible, but he was the first to put the notion on a sound and testable physical basis. Thus Hutton is considered the father of deep time, although he never used the phrase.
A century later, the age of the Earth was widely considered to be some tens or hundreds of million of years. There was little hard evidence to constrain speculation until the discovery of radioactivity and 20th-century advances in physics that brought about radiometric methods of dating rocks. By the mid-1900s it was clear that Earth was about 4 billion years old, more than enough time for all the geologic history we could envision.
The term "deep time" was one of John McPhee's most powerful phrases in a very good book, Basin and Range, first published in 1981. It first came up on page 29: "Numbers do not seem to work well with regard to deep time. Any number above a couple of thousand yearsfifty thousand, fifty millionwill with nearly equal effect awe the imagination to the point of paralysis." Artists and teachers have made efforts to make the concept of a million years accessible to the imagination, but it's hard to say that they induce enlightenment rather than McPhee's paralysis.
Geologists do not talk about deep time, except maybe rhetorically or in teaching; instead they live in it. They have their esoteric time scale, which they use as readily as we talk about our neighborhood streets. They use large numbers of years nimbly, abbreviating "million years" as "myr." In speaking, they commonly don't even say the units, referring to events with bare numbers. Yet it's clear to me, after a lifetime immersed in the field, that even geologists can't really grasp geologic time. Instead they have cultivated a sense of the deep present, a peculiar detachment in which it is possible for the effects of once-in-a-thousand-year events to be seen in today's landscape and for the prospect of rare and long-forgotten events to occur today.