The basic concept of Keynesianism is that each economy has a level of [aggregate demand], which does not respond to price or income levels in the same way that [classical economics] says it should. Rising income, for example, does not lead to a matching increase in consumption or business investment. Business investment is driven by investment opportunity, not {only by interest rates. Savings is driven by [liquidity preference], not only by interest rates.
Keynes suggested that, for any economy, there was a [marginal propensity to consume] that was less than one. Hence, if the national income rose by 10%, consumption would rise by something less than 10%. This would lead to some production not being consumed, waste, and unemployment.
What Keynesianism Says We Should Do

In 1936, when Keynes wrote The General Theory, most of the world was suffering from the Great Depression. Keynes recommended that the national government stimulation aggregate demand through a policy of deficit stimulus. In other words, the country should create adequate levels of aggregate demand by spending more than it took in as taxes (fiscal policy).
Also, Keynesianism held that aggregate demand could be stimulated up to a point by lowering interest rates (monetary policy).
Application

In the USA and other large industrial countries, fiscal and monetary policy has been attempted often. After 1980, the Federal Reserve chair (Paul Volcker) was a monetarist, who claimed to reject Keynesianism. Nobel laureates in economics almost unanimously attacked Keynesianism as outmoded and wrong-headed, but governments continue to use fiscal stimulus and interest rate cuts in response to recessions.