We are currently in the following:
• The Phanerozoic Eon which began 542 million years ago. Phanerozoic means "evident life" which refers to the Cambrian Explosion, a time when life diversified and, more importantly, developed hard parts (like shells) which were readily preserved in the fossil record.
• The Cenozoic Era which began 65.5 million years ago at the time of the demise of the dinosaurs. The three Eras of the Phanerozoic Eon are Paleozoic ("ancient life"), Mesozoic ("middle life"), and Cenozoic ("new life").
• The Quaternary Period which began 2.6 million years ago when the most recent Ice Age kicked off. Quaternary means "fourth" and refers to an old four-fold division of geologic time no longer used.
• The Holocene Epoch which began only 12,000 years ago with the retreat of the last great ice sheets. The name derives from two Greek words meaning "entire" and "new" and refers to modern times. The Holocene is not subdivided into Ages since it's so short.
Some scientists are now advocating for a new division of geologic time - the Anthropocene Epoch from the Greek word for "man" - ἄνθρωπος (anthropos).
According to Wikipedia, the term Anthropocene was first coined by University of Michigan professor of biology Eugene Stoermer (1934-2012) in the early 1980s but didn't get much notice until it was popularized by Dutch Nobel prize winning atmospheric chemist Paul Crutzen in International Geosphere-Biosphere Programme Newsletter 41 (2000) where he wrote, along with Stoermer:
To assign a more specific date to the onset of the "anthropocene" seems somewhat arbitrary, but we propose the latter part of the 18th century, although we are aware that alternative proposals can be made (some may even want to include the entire holocene). However, we choose this date because, during the past two centuries, the global effects of human activities have become clearly noticeable. This is the period when data retrieved from glacial ice cores show the beginning of a growth in the atmospheric concentrations of several "greenhouse gases", in particular CO2 and CH4. Such a starting date also coincides with James Watt's invention of the steam engine in 1784.
In other words, they date the start of the Anthropocene to the Industrial Revolution. Others, like retired University of Virginia paleoclimatologist William Ruddiman have proposed a start date for the Anthropocene around 6,000 BCE to coincide with the earliest movements from a hunter-gatherer to an agrarian society
Anyway, the basic idea of the Anthropocene is that man (and woman, let's not be sexist) has altered the Earth enough to make a mark in the geologic record. It's become a fashionable term, even the Geological Society of America titled their 2011 annual meeting Archean to Anthropocene: The past is the key to the future.
Much of the advocacy for the term Anthropocene seems to come from environmentalists. It's clearly a catchy term, is used by an increasing number of scientists and scientific organizations, and fosters awareness of humanity's effect on the natural world around us (usually a negative effect).
There are some dissenters, however. A recent article in GSA Today (the newsletter of the Geological Society of America) titled "Is the Anthropocene an issue of stratigraphy or pop culture?" by Autin and Holbrook, argues that the term may not be all that useful as a formal stratigraphic term for geologists.
The boundaries of the geologic time scale have some reality in that they're based on some type of signature in the rock record - typically fossils but they can be lithologic (rock type), mineral, chemical, or geophysical signatures. At these places, we place what's metaphorically called a "golden spike" to mark that boundary. Technically, it's called a Global Boundary Stratotype Section and Point (GSSP).
Here's a bronze plaque marking the GSSP for the base of the Ediacaran Period (630 million years ago) which is located in the bed of Enorama Creek in Brachina Gorge in the Flinders Ranges of South Australia (31° 19' 53.8" S, 138° 38' 0.1" E).
So, Autin and Holbrook argue, where should we put that "golden spike" for the base of the Anthropocene? While it's a neat term with pop culture appeal, how exactly are we going to define this scientifically? While humans leave a lot of traces on the Earth - cities and roads, Earth works, changes in soils, not-biodegradable materials, mining of Earth materials, extinctions of plants and animals, alteration of ecosystems, radiaoactive signatures in the environment, chemical signatures in the environment, changes in atmospheric chemistry and climate, etc. - one has to imagine how these things might or might not leave a lasting impression in the stratigraphic record of the future.
Therefore, while Anthropocene may survive as a pop culture term or a concept in environmental science, it may not be appropriate as a formal geologic time scale Epoch.
Here's an article titled Enter the Anthropocene - Age of Man from National Geographic if you'd like to read more.