The Neogene is the geologic period that began 23 million years ago and extends until 2.6 million years ago or until the present day, depending on which scientist you ask. Because it occurred relatively recently or is still ongoing, the organisms of the Neogene tend to be mostly similar to today's, but with many important differences. Because humans have caused the extinction of thousands of species since we began spreading across the world 60,000 years ago, the Neogene historically had more species than we observe today.
During most of the Neogene, the continents were mostly in their present-day positions, though South America was disconnected from other continents and did not connect with North America until 3 million years ago. South America had among the most unusual fauna on the planet, dominated by marsupials, including large carnivorous marsupials, xenarthrans (armadillos, anteaters, and sloths, including the elephant-sized Megatherium), and a unique and diverse group of native ungulates, most of which went extinct when animals from North America traveled south and outcompeted the native fauna.
The Neogene period is characterized by an ongoing Ice Age, which began at approximately the start of the period and continues to the present day. It has been punctuated by glacials and interglacials, periods where continental glaciers cover much of North America and Eurasia and other periods when they recede to Greenland and Antarctica. In contrast to the warmer, wet periods before, the Neogene is characterized by increasing aridity and the replacement of forests with grasslands. Thus, many of the characteristic Neogene organisms are ruminants -- animals that chew a cud, an adaptation to extracting more energy from grass.
While today, large animals are relatively rare, they were much more common during most of the Neogene. Eurasia was colonized by numerous mammoths, including the wooly mammoth. Some parts of Europe and the Near East were colonized by animals traditionally associated with Africa, such as hippos, water buffalo, lions, and cheetahs. Much of the caves of Eurasia were filled with hostile animals such as the cave bear or cave hyena. It is thought that competition between humans and cave hyenas slowed down our migrations across the Bering strait by thousands of years.