When watching your aquarium fish, it's not uncommon to see one that appears to be hovering in one spot. Don't worry if a fish seems to be taking a break from swimming; most fish won't drown if they stop swimming. Water needs to touch their gills so they can breathe, but most fish can move water through their gills by opening and closing them. The swimming movement isn't what keeps most fish from drowning, but a few must swim to breathe.
Fish tend to be strong swimmers with plenty of stamina, but they needs breaks occasionally. Many fish have daily cycles, just like land animals do -- some are active during the day and sleep at night, while others take the days off and move around in the dark. As most fish don't have eyelids, they can't close their eyes, but they become stationary, sink to the bottom or ride a pleasant current while they rest. An aquarium isn't likely to have those kind of currents, but your fish might look for an out-of-the-way spot to take a break. They are still breathing, not drowning; most fish don't need water to flow across their gills, but the gills must be in contact with the water.
How Gills Work
Gills perform oxygen exchange for fish, drawing air out of the water. Gills typically are full of tiny blood vessels that are close to the surface to help with this gas exchange, releasing carbon dioxide and breathing in the oxygen. You'll often see fish gills opening and closing, even when the fish are sitting still, which brings the gills in contact with the water so the fish can breathe.
Sharks are a notable example of fish who risk drowning if they stop swimming. Shark gills are slightly different than regular fish gills. Their gills are designed to require moving water to breathe; the water must flow across the gills' surface to allow for gas exchange. If a shark isn't swimming, the water isn't flowing over his gills and he can't breathe. This is true of a few other fish species, including Atlantic bluefin tuna, although these fish, who often top 6 feet long, won't fit in most home aquariums.
Signs of Trouble
Although your fish are unlikely to drown if they stop swimming, a reduction of normal activity could signal a problem. Many fish slow down or stop swimming when they are sick. If you notice a fish is less active than normal, check for other signs of disease, such as a change in skin color, fuzzy growth on his skin or dull eyes. Changing the water might help reinvigorate your fish, or he might need medication such as antibiotics. Separate sick fish from the rest of your aquarium creatures as quickly as possible to prevent the possible spread of infection.