The Oligocene is an epoch of the Paleogene period and the Cenozoic era, extending from 33.9 to 23.0 million years ago. The Oligocene marked the start of the cooling trend that would continue for the rest of the Cenozoic, until today. Continental glaciers first began to form on Antarctica, and tropical broadleaf forests which had extended 45 degrees from the Poles began to contract more closely to the equator, while coniferous forests, which had previously extended to the Poles, began to recede as well. The Oligocene was preceded by the Eocene and followed by the Miocene.
Oligocene sea levels were higher than today's, on account of less fresh water being frozen up in glaciers, leading much of central Eurasia, the southeast United States, and southeast Asia to be flooded. North and South America were separated by the Panama Strait. Though the higher sea level suggests that North America and Eurasia would have been separated, there must have been a land bridge between them during the period, as the faunas of these two continents were so similar.
Grasses, which began their career in the late Cretaceous (~85 mya) as plants growing on water boundaries, were finally beginning to dominate over the previously planet-wide forests, growing in open areas and challenging animals to evolve from arboreal to grassland-type adaptations. Among the most prominent of these was the evolution of the four-chambered stomach and the practice of rumination (chewing a cud), which in effect transformed the stomach into a fermentation vat to extract nutrients from tough grass.
Many of the "normal-looking" mammalian faunas which we are so familiar with today began to evolve during the Oligocene. Deer first evolved in the early Oligocene. Carnivorans during the period were primarily canid -- bears and dogs -- as felids did not evolve until just after the end of the Oligocene, about 20 million years ago. The first fully recognizable cetaceans (whales, dolphins, etc) evolved a few million years before the start of the Oligocene and were progressively growing in size.
"Unusual" animals, including predators, still existed in the Oligocene, but many of them started to phase out. These include the mesonychids, large hoofed predators with oversized skulls that ruled the Asian steppes and the North American plains, but they were on the decline by the Oligocene and died out early on in the period. Phorusrhacids -- "terror birds" -- were the dominant predators in South America, which they had been since early on in the Cenozoic and continued to be until South America connected to North America and competitive mammalian predators started taking over their space. Primates during the Oligocene were mostly primitive, similar to modern-day monkeys, but apes did evolve in Africa at the very end of the period.