What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series? Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar sounding names). I'm hoping you'd come up with an explanation that is so intuitive and mathematically beautiful that they'll get embedded into my memory forever and this thread will serve as a super quick refresher whenever I (or anyone else) needs it.
I wrote this handout as a complement to Oppenheim and Willsky. Please take a look at Table 4.1 on page 14, reproduced below. (Click for larger image.) I wrote that table specifically to answer questions such as yours. 



Note the similarities and differences among the four operations:


"Series": periodic in time, discrete in frequency 
"Transform": aperiodic in time, continuous in frequency 
"Continuous Time": continuous in time, aperiodic in frequency 
"Discrete Time": discrete in time, periodic in frequency


I hope you find these notes helpful! Please feel free to distribute as you wish.