Why are the sodium ions not reduced to sodium metal during the electrolysis of brine?
Because hydrogen is less reactive than sodium
Brine is a concentrated salt solution, meaning that it is a liquid with lots of free ions in it. We have the ions from the salt,
The positive ions (Hydrogen and sodium in this case) are attracted to the negative electrode (the cathode) due to having opposite charges. However, there is a rule at the negative electrode that the least reactive metal will be discharged (turned into atoms). Because hydrogen is less reactive than sodium, the hydrogen is discharged as
To understand this, you can think of a reactive atom as really wanting to be an ion - they almost have full outer shells and so they really want to gain those few extra electrons (talking about a positive ion here). A less reactive atom isn't too fussed either way. So, if we have hydrogen and sodium ions in solution, the reactive sodium really wants to stay as an ion, and the less reactive hydrogen isn't too fussed about it so it takes one for the team and gains some electrons.
Obviously, this is a simplified version, but it sort of creeps into quantum physics stuff...
Hope this helps; let me know if I can do anything else:)