Hi Elva V.,
To some extent, that might depend on concentrations -- very weak solutions behave like water. But usually, we are considering substantial concentrations. SO -- at the reducing electrode, Na(+) is far less reducable than is H+, so H2 will be released (you know this, because Na(0) + H2O = Na(+) + OH(-) + H2 is spontaneous). But at the oxidizing electrode, Br(-) is more oxidizable than is OH(-), so Br2 is formed instead of O2 .
The idea of relative strengths of oxidizers and reducers is a pervasive one, and lots of commercial chemistry is based on it. For example, Cl2 gas + a bromide-containing brine -> Br2 + chloride is spontaneous, and a quick way to strip the bromine out (it's valuable!, and preferred for certain pool-sanitizing conditions, even if more expensive).
--Cheers, --Mr. d.