Floods refers to body of water covering area which is normally dry land. Floods may occur along the river banks, lakes and sea coast. However, river floods are most common.
River floods and lake floods occur because of too much water flowing into them, that overflows its banks, or sometimes even breaks the banks. The excess water is the result of excessive rains in the catchment area or/and sudden melting of snow and ice. Sea floods are caused by unusually strong sea waves hitting the coast. This may happen at the time of cyclonic storms or by tsunami waves. Sea floods caused by tsunami can be particularly sudden and violent.
In general, storms cause damage to agriculture, people and property. Floods damage standing agricultural crops and may also carry away the top soil making the land barren. Floods damage all immovable properties that get submerged in flood waters. Any other property which cannot be removed to safer places during floods are also damaged. When the storms are sudden and extensive the damage is more because of difficulty of removing property to safer places. In such cases people may also be unable to move to safer places and may drown.
The only good associated with floods is that sometimes it leaves behind fertile soil on the land after the floods. For example, the yearly floods in Nile have made plains of Egypt one of the most fertile land in the World.
Your other definition for floods is more than accurate. As far as it's effect on agriculture; it depends on the type of flooding. In El Paso, Texas (an arid desert region), flooding is used as watering tool to promote deep roots. The areas planted are flooded with water that get about 2-4 inches deep every 3 weeks or so. The water would be allowed on the area needing water for a specified amount of time through cannals. Then, it would be cut off and the water would be allowed to dry up over a period of about a week. The crops with no flash flooding are great and allow an arid region to produce a product it would normally not be avaliable to do.