What is the role of complex exponential?
Euler's relationship says that $e^{j\Theta}$ is equal to $cos(\Theta) + j*sin(\Theta)$.  The Fourier Transform can then be seen as correlating the signal with sinusoids at various frequencies.  The continuous Fourier Transform correlates with an infinite number of sinusoids, while the discrete transform uses $N$ sinusoids, where $N$ is the length of the transform.