The oldest living species on Earth are called cyanobacteria (blue green algae). Evidence shows that the cyanobacteria have been around for nearly 4 billion years, right around the same time any type of life began in the first place. While they seem to be very good at survival, they are quite poor at evolution, but maybe that's why they've been around for 4 billion years. The evidence that points to the idea that cyanobacteria existed 4 billion years ago is found in other fossils from around that time period. In both current day samples and ancient samples the cyanobacteria have been found. If the cyanobacteria are actually not the oldest species on Earth, it would surly be some other type of bacteria, such as stromatolites, which are mounds of algae and bacteria.
Learning about bacteria perhaps was not your intention when you asked this question so I'll also note that the oldest animal on the planet is the brachiopod Lingula, which has been around for over 500 million years. The Horseshoe crab has been around for a few hundred million years as well.
Here's are pictures of the blue green algae and the Lingula.
Algae
Lingula
