Operators and Enhancers/Silencers
Enhancers and silencers are binding sequences for transcriptional activators or repressors, in which case the sequence is often located some distance upstream or downstream of the gene it regulates. See regulation of transcription for information about how these interact with their target genes (through DNA bending, mediator, etc.). A note about enhancers and silencers, though, they're not necessarily required for transcription: they help the gene attain robust up- or down-regulation of transcription.

The operator, on the other hand, influences whether the promoter will do something, or nothing. If we're talking about inducible systems, the repressor is bound to the operator, blocking action to/from the promoter (an inducer will bind the repressor and keep the operator from blocking the promoter). If we're talking about a repressible system, the repressor will need a corepressor to bind the operator to abrogate transcription by blocking action to/from the promoter.