Omnivores are named due to the Latin prefix omni, which means "all." For lack of a more descriptive qualifier, it means omnivores eat everything! Of course, that's not completely accurate, but it does mean an animal that eats both plant matter and flesh. The difficulty in classifying an animal as an omnivore has a number of facets.
First, many animal species have different periods of development when it will transition from eating one type of product to another. For example, a cow will drink milk (an animal product) while it is a calf, but then transition to grass and grain once it weans. Many insects also ingest flesh products as larva, then live on plant matter once they mature.
Another classification issue in labeling the type of diet is that there are species in which the males eat one source and the females eat another, such as some common mosquitoes in which the male eats plant matter and the females drink animal blood to acquire the nutrients necessary to reproduce.
Opportunistic feeding is another problem when trying to classify animals according to their diets as either omnivore, herbivore, or carnivore, as most species will eat what their environments provide. In times of lower availability, even obligate carnivores like cats will eat grains and plants, while omnivores will often have to stick to either plants or animals during different times of the year when one food source is less plentiful.
Within the bear family, there are different species of bears that are either obligate carnivores (strictly meat eaters) or omnivores. Polar bears, for example, live on a diet of strictly meat sources, possibly due to the scarcity of vegetation in their environments. Other bears, such as grizzly bears or brown bears, still eat meat and fish sources but also live on berries, plants, and vegetation.
While humans can make the personal decision for health, political, or spiritual reasons to become a vegetarian (eating only plant matter and animal byproducts like eggs or milk) or a vegan (no meat sources or animal products), their purposeful decision is not a physiological requirement. Human bodies are still built to ingest and absorb nutrients from both plant sources and animal sources, regardless of what individuals choose to consume.
Birds present an interesting twist in classifying them as omnivores, herbivores, or carnivores. As it turns out, birds that live on seeds and grains as adults often consume insects as newly hatched birds; even adult members of the species that feed themselves on plant matter will often feed their offspring insects. Nectar-feeding birds such as those that draw nutrients from flowers and fruits will actually consume the ants and other insects that are hidden inside the flower, drawing nutrition from these animal sources.