Question about Euler condition In dynamic macroeconomic model(without production sector), Euler equation is $$U'(C_t)=b(1+r)U'(C_{t+1})$$    

I found another equation related with Euler which is called Euler condition.
In New Keynesian model, 
$$Y_t-E(Y_{t+1})=-\left(\frac{1}{\sigma}\right)(i_t-E(\pi_{t+1})-\rho)$$ 

which means Euler condition. In this equation, what is the Economic meaning for Euler condition?
The Euler Equation typically refers to the interior optimal choice between consumption today and tomorrow (or some similar intertemporal choice).  That is, it equalizes the marginal utility of consuming a unit today vs saving that unit in order to consume tomorrow.  This yields the first equation you listed (where $\beta=\frac{1}{1+\rho}$):

$$
u'(C_{t}) = \frac{1+r}{1+\rho}u'(C_{t+1})
$$

I have always referred to the second equation you describe as the "New Keynesian IS Curve" as in Gali's textbook and several NK model papers.  But it does come from  the same intertemporal consumption choice Euler Equation.  

Based on the equation you've provided, my guess is the model assumes a few things:


CRRA utility of the form:  $ u(C) = \frac{C^{1-\sigma}}{(1-\sigma)}$
No capital investment such that $ Y_{t} = C_{t} $


Using these assumptions, the Fisher Equation ($ r_{t} = i_{t} - E_{t}[\pi_{t+1}] $), and the Euler Equation you gave above, you would arrive at what you're calling the NK Model's Euler Equation. (Note that your second equation is also probably in log terms)

At the end of the day, the two are the same thing: The first order condition that is making consumption equivalent to saving in terms of utility.  The latter version just tacks on additional features of the NK model to arrive at an equation that's useful for describing aggregate dynamics (along with the Phillips Curve and a monetary rule).