All time-domain sampled signals exist as finite-length, discretely-sampled sequences. The Fourier transform is mathematically defined as going from -Inf to +Inf on a continuous signal, clearly impossible with a sampled signal.
Two distinct effects appear with sampled signals that do not appear with continuous signals, the first is the Nyquist frequency and the second is the Gibbs phenomenon.
The sampling process creates an upper limit on the frequencies used to approximate it (called the Nyquist frequency) that is one-half the sampling frequency. This is the highest frequency that the sampling process for that signal can uniquely identify. Hardware analogue-to-digital converters (ADCs) incorporate a hardware Bessel filter to filter out all frequencies above the Nyquist frequency to avoid ‘aliasing’. Frequencies higher than the Nyquist frequency otherwise would be sampled, and appear as low-frequency energy that are not actually present in the sampled signal.
The truncation of an infinite-length signal to a finite sampled length creates another problem at the ends and at any other abrupt transitions known as the Gibbs phenomenon. This is due to the inability of a finite series of frequencies (dictated by the Nyquist frequency limit) to closely approximate the beginning and end of the sequence and any abrupt transitions within it. This results in a ‘ringing’ effect, the Gibbs phenomenon, most noticeable on older JPEG photographs (that use a discrete cosine transform compression scheme) as ‘ghosting’ around the outlines of abrupt transitions.
The windows of various types convolve (in frequency or time domains, depending on the application) with the discrete Fourier transform of the signal (or in the time domain) to reduce the effect of the Gibbs phenomenon, especially at the beginning and end of the sampled sequence. This is especially important in filter design, where the frequency content is reduced even further than in the original signal.
This is a long and somewhat indirect approach to your Question. It requires a reasonable explanation of both effects because they interact. A full description and mathematical derivation with examples of the approximation of a continuous signal of infinite length with a sampled signal of finite length can be found in any comprehensive textbook on digital (discrete) signal processing. I will not attempt to reproduce it here.
I invite others with more sophisticated signal processing knowledge and experience than I have to add anything I omitted that could be significant, and to correct any errors.
