Basic New Keynesian Model: Why variables are defined by integral equations In the basic New Keynesian Model most variables are described as an integral equation. Lets for Instance look at labour demand which is defined as:
$$N_t=\int_{0}^{1}N_t(i)di$$

What is the intuition behind that. How does the $N_t(x)$ function look like and why do we integrate it interval $[0,1]$?

I am reading Gali's book.
First of all, this is not something special to "the basic New Keynesian Model". It is a commonly used technique when modeling an economy with competitive environment.

In your case, the author wrote that there is a continuum of differentiated goods represented by the interval [0,1] and each firm is producing one of the goods. You can imagine that there are uncountably many different kinds of good in the market. As a result, the size of each firm is negligible (just a point on the [0,1] interval) compared to the entire economy. Each firm indexed by $i$ solves its individual profit maximization problem to obtain its labor demand $N(i)$. Then, the aggregate labor demand is simply the "sum". In this case, the sum is replaced by the integral because the variety of different firms is the whole unit interval.