Correlative connectors consist of two coupled connectors, one in each sentence:
 
- BOTH . . . AND  (= and)    both are true  [ + + ]
      She both plays the piano and sings  
      she plays both the piano and the guitar 
 
- NEITHER . . . NOR    both are untrue  [ - - ]     /naɪðə ... nɔ:/ (American: /ni:ðər ,,, nɔ:r/)
      Neither James nor Virginia was at home  (also: were)
      I neither smoke nor drink
 
- EITHER . . . OR  (= or)    only one can be true  [ + - ] , [ - + ]     /aɪðə ... ɔ:/ (American: /i:ðər ,,, ɔ:r/) 
      You can either come with me or walk home
      either you leave this house or I'll call the police
 
- NOT ONLY . . . BUT ALSO   both are true  [  + +  ]  (formal)
      She not only sings like an angel, but also dances like a nymph

Something special about these is that if NOT ONLY is at the beginning of the sentence, we need an inversion:
      Not only does she sing like an angel, but also dances like a nymph

BALANCE
 
These correlative constructions must be "balanced", that is, the structure of each part of the sentence must be identical. Unbalanced constructions are not grammatically wrong, but they are considered to be very clumsy.
 
- She both plays the piano and the guitar

This sentence is unbalanced (and so, not appropriate for a composition) because the structure is: (both + verb)  (and + object)
So, to make it balanced we have two possibilities:
 
- She both plays the piano and plays the guitar  (both + verb / and + verb)
- She plays both the piano and the guitar  (both + noun / and + noun)
 
The second sentence is much better because repetitions (plays....plays) are very clumsy and not elegant.

For Spanish speakers

- BOTH . . . AND  (= tanto ... como / y además) 
- NEITHER . . . NOR  (= ni ... ni)
- EITHER . . . OR  (= o bien ... o bien)
- NOT ONLY . . . BUT ALSO  (= no sólo ... sino también/sino que además)