A "native speaker of English" refers to someone who has learned and used English from early childhood. It does not necessarily mean that it is the speaker's only language, but it means it is and has been the primary means of concept formation and communication. It means having lived in a truly English-speaking culture during one's formative years, so that English has been absorbed effortlessly as by osmosis.
Literally, a native speaker of English is somebody that learned English as their first language.
In reality, I would say that a native speaker of English is a speaker of English which also thinks primarily in English and which other native speakers of English would recognize as such. It's certainly a circular definition but I think that that's a key part of any definition. Trying to leave it out caused me to give an incomplete definition earlier.