The National Weather Service is the only agency that issues official weather advisories including watches and warnings.
A watch is used when conditions are favorable for dangerous weather to occur. The weather is not currently occurring.
A warning means dangerous weather is happening in a certain region or is extremely likely to happen soon.
Weather watches typically come many hours before warnings. For example, a winter storm watch is usually declared for an area from 12 to 36 hours prior to a storm, while a warning indicates that a storm has been observed nearby and is about to hit the area.
Another type of weather alert is an advisory. An advisory is similar to a warning except that the weather conditions are not as severe.