An alluvial plain is a flat, fan-shaped landform created by the long-term deposition of sediment by a river. Alluvial soil deposits form after repeated flooding of the area; an alluvial plain is usually formed from multiple, shifting floodplains over a very long period of time, from decades to centuries to millennia.
As a river flows it carries sediment, which settles to the bottom of the river. This slowly raises the river, leading to ever-increasing chance of flooding its banks. Meanwhile, rivers form meanders as a flow cuts into one bank (the more concave bank, all things being equal), depositing more sediment on the convex bank; over time this may form oxbows. Between flood stages and meanders, a river can end up impacting a very large area over geologic time scales.
As these plains tend to be well hydrated, they are often wetlands such as swamps, mangroves and mudflats. When drained or cleared, the large, flat, plains of fine soil and organic deposits tend to make for good farming.
A smaller alluvial 'plain' formed by streams (generally in a mountainous region) is known as an alluvial fan. A related land form is the river delta, in which a river empties into a larger body of water, resulting in a lower rate of flow and a deposition of sediment in a wide fan.