They are use when distinguishing between 2 things. For more than two things, we would use 'any' or 'none'. They work in the same way as these words but are specifically used for 2 things.
either is the positive form and it means 'one or the other'
You can have either tea or coffee.
neither is the negative form (not either)
Neither John nor Susan went to the conference. [notice we use 'nor' rather than 'or]
So John didn't go and Susan didn't go. [John didn't go and nether did Susan.]
Either John or Susan can go to the conference.
This means that one or the other can go - both options are suitable.
You should also compare the word 'both' with these words (this would be all when compared to any and none).
Both Susan and John went to the conference. [Susan went and John went.]
Any of our team can go to the conference. [Anyone can go.]
None of our team went to the conference. (No one went.]
All of our team went to the conference. [Everyone went.]