Ants belong to the order hymenoptera, so named for the membranelike pairs of wings each species sports. The four common hymenopterans, ants, wasps, bees and sawflies, share a common body type, although bees are so broadly built, they would be difficult to mistake for ants. Like all insects, they have segmented bodies divided into head, thorax and abdomen segments. In hymenoptera, the segments are usually distinct, and the abdomen is often slightly elongated and larger than the thorax.
The ants most commonly seen by humans are female workers who have not sexually matured. These ants never mature or grow wings; their sole job is to help establish and defend the colony. Once a colony is established, however, many winged, sexually mature males, and a few females will be born. These winged ants seek out a high place during the very brief mating season. So if you see an insect that resembles a big, black ant with wings, it may just be a big, black ant with wings.
If it's not an ant you're seeing, the next likely culprit is a wasp. Approximately 30,000 species of wasp have been identified, and several are large and black. There are many wasps that do not have stingers, and among those that do, stingers, which are modified ovipositors, are present only on the females. It would be very easy to confuse a black wasp, particularly one with no stinger, with a large flying ant.
Sawflies are hymenoptera that strongly resemble wasps in their adult stages and are often referred to as wood wasps and are often grouped with wasps. Like other hymenopterans, as adults they have long, enlarged abdomens and two pairs of thin, membrane wings. Though their larval stages resemble something more like a caterpillar, as adults, they also resemble large, winged ants or wasps.
Outside the order hymenoptera, there are thousands of insects, and even spiders, that have evolved to mimic the appearance of ants. This mimicry of ants is known as myrmecomorphy. Ants are mean and purportedly foul-tasting, so ant mimicry is a good way to deter predators. There are no winged spiders, but many myrmecomorphic insects retain their wings.