Gel electrophoresis is kind of a general term for several associated techniques but by itself, it can be used to separate nucleic acids or proteins based on size and other factors. Usually, you denature the nucleic acids (with urea) or proteins (with SDS). But you can run it without SDS and it's then called native PAGE, which separates proteins based on not only size but also 3D structure and charge. You also usually run it under reducing conditions (you add a reducing agent) so that the disulfide bridges are broken and you get full extended form. But sometimes you might want to keep the disulfides - like if a disulfide holds two subunits together and you want to figure out what the weights of those subunits are, you might do two trials - one reducing and one not - in order to figure that out.
Western blots are done on proteins. After you run a gel electrophoresis, you transfer the protein onto a membrane and then you "blot" it with an antibody for that protein. Then you incubate with a secondary antibody that you can detect (fluorescently, etc.) that binds to the first antibody. Southern and northern blots operate under similar scientific principles although the details are slightly different. Southern blots detect DNA and northern blots detect mRNA. In both of these methods, you separate the nucleic acids with electrophoresis and then treat with a probe that is complementary in sequence to the one you're looking for. That probe also contains a reporter of some sort that will allow you to detect it after you've treated the gel/membrane.
I'll let a biologist talk about the specifics of the difference between traditional PCR and RT-PCR but the PCR method is generally used to amplify some segment of DNA. You have some sequence of DNA you want lots of copies of, so you make forward and reverse primers that will anneal to the two ends of the gene on opposite strands. Then you basically cycle heating and cooling - it used to be done manually by graduate students but now it's all automated. And a few hours later, voila - you have lots of copies of what you want.
Isoelectric focusing is used with proteins. Each protein will have an isoelectric point and so this takes advantage of that by basically running a protein along a pH gradient until it stops moving.
In most of these, you'll want your molecules to have no 3D structure - you want to denature everything so that it runs according to size on the gel and so that your probes can bind to their respective sequences. Secondary and tertiary structure is also more important for proteins than with nucleic acids because structure and function are so closely related in proteins and not as much so in nucleic acids. So sometimes you'll want to figure out something to do with tertiary structure with proteins and that's when you start to do native or non-reducing gel electrophoreses to parse those out.
