Swimmers can move in water without drowning.
Dabbling or untrained swimmers will start drowning immediately upon contact with water, and require a ramp or stairway to get out. If neither are accessible, they're done for. Unfortunately outdoors natural pools and ponds do not have ramp edges. One can channel a safety ramp into each pond however.
A novice swimmer is able to safely get out of water without needing a ramp or stairway, but they will start drowning if stunned, and once that happens it can be difficult to get them out, as they lose the ability to exit anywhere and behave just like an untrained or dabbling swimmer. Namely, they start drowning. Everyone is stunned by falling into water rather than entering it calmly, which is what normally happens when they aren't entering it of their own free will.
Adequate swimmers do not panic and start drowning like novices do in that sort of situation, even when attacked, so training to this level is highly recommended. Higher levels only increase the speed when swimming with a legendary swimmer being faster than he would be while running.
 Learning/Teaching swimming
A dwarf in the water will gain the ability to swim very fast - sadly not fast enough to prevent dying from drowning. While water with a depth of 7/7 is deadly for non-swimmers, less than it will not harm any dwarf. So you can use water from 4/7 to 6/7 safely to teach your little ones how to swim. The speed of learning is independent from the depth, but water with a depth less of than 4/7 is not deep enough to make a dwarf swim and so to learn.
Training your little ones just requires a place of constantly or temporarily 4-6/7 water. Military orders or making rooms a meeting hall will not entice dwarves in to the water, so you may need to prevent them from leaving an area (locked door etc) and then fill the area with the required amount of water, or dump them in from the above z-level using a draw bridge.
A fully automated method to train idlers is to use water flowing over a 1-z drop, with a 1-wide meeting zone at the top of the ledge, and a swimming pool at the bottom. Idlers will go to the meeting zone, be swept over the side into the pool and swim to the ramp, and repeat this for as long as they are idle.
Swimming, since it involves no activity, can be potentially useful to train physically handicapped dwarves, whose conditions might go away or become manageable with an attribute boost to strength, endurance, willpower etc.
 Adventure Mode
In adventurer mode, as a novice swimmer, by moving carefully ( +direction) into open space above water and selecting the option to move below (such as West/Below), then you can swim about without getting stunned and starting to drown. To get out, alt-move carefully against a shoreline and select the option to move above.
Swimmers can also dive and rise through the z-axis by pressing and respectively. Note that air-breathers will be unable to breathe without air in the tile above them, and without returning to the surface will eventually drown. (Sadly, there's no oxygen meter as of yet, so you'll never know when they're about to expire. Don't linger too long.)
In adventure mode, water preference can be switched between "when possible" and "necessary" by pressing.
If the temperature (press) is "freezing" or if it is "cold" and close to sundown, the water may freeze while you are swimming, which instantly kills you and leaves your your frozen corpse encased in ice and a valuable find for archeologists.