I understand that the time domain representation of white noise looks like impulses. How do colored noises like brown, pink etc. look like when we perform an inverse Fourier Transform on them ? What could be some sources of colored noise that may affect speech signals?
I guess you ask how they sound rather than how they look, right?
The "color of the noise" corresponds to the perception you would have from mixing different perfectly chromatic wavelengths: white if you cover all frequencies, pink when there is less blue (a smooth drop in high frequencies), brown (if the drop is even more sharp) and so on...
Similarly, the "color of the your noise" in the temporal domain should be put in analogy to natural sources of noises:
White noise implies no correlation between samples of the noise, even consecutive samples. Colored noise, therefore, implies that there is correlation of some sort between the noise samples, which in turn implies that we can take advantage of that correlation to get rid of some of the noise.
Beyond that, there is not a lot that we can say about what it looks like in the time domain other than to say that colors that favor low frequencies, like pink noise, will tend to change more slowly, while high frequency colors, like "blue" noise, will tend to change more quickly.