The Euler equation relates time preferences and real interest rates to the decision of whether to consume today or tomorrow/next year/next period. Time preferences indicate how "patient" you are, since money/consumption now is worth more to you than money/consumption later is. And real interest rates indicate the rewards for being patient, since higher real interest rates mean that deferring consumption now gives you greater income to consume later. When these two perfectly match up, then consumption should be steady over time, and consumption should either be perfectly stable (in the event where nothing random ever happens) or should follow a random walk (in the event where there are random, uncorrelated shocks to income at different points in time). If you raise (lower) the real interest rate, then you have also raised (lowered) the growth rate of consumption.

The Euler equation is pretty foundational to a lot of modern macroeconomics, since it's one of the best ways they have of modeling the consumption decisions of forward-looking agents in a way that satisfies the Lucas Critique. It can also have pretty big implications for the role of monetary policy. If you assume that the economy trends towards its potential (a standard, if maybe dangerous thing to do), then increasing the growth rate of consumption means lowering the level of consumption today, since we're assuming that future consumption is (eventually) fixed by the potential of the economy. This gives us the standard New Keynesian policy prescription that higher interest rates are contractionary and lower interest rates are expansionary, all without needing to use either explicitly monetarist reasoning (by talking about the quantity of money) or explicitly Old Keynesian reasoning (by talking about IS-LM).

However, the Euler equation might be a pretty inaccurate microfoundation to use, and I'm currently in a massive personal crisis w.r.t New Keynesianism and the Euler equation and am still waiting for /u/Integralds to finish assuring me that everything will be okay. So take this all with a grain of salt.