Carbohydrates: Reducing Sugars

A reducing sugar has a free (or potentially free) aldehyde or ketone group, so it can reduce Tollens' reagent and Fehling's solution.
Galactose is a monosaccharide (an aldohexose). In solution its ring opens to give a free aldehyde group, so it is a reducing sugar.
Gluconic acid is formed by oxidation of the aldehyde group of glucose to –COOH, so it no longer has a free aldehyde group.
In $\beta$-methyl galactoside, the anomeric –OH is converted into a –OCH$_3$ group (a glycoside), so the ring cannot open; it is non-reducing.
In sucrose, the anomeric carbons of glucose and fructose are joined to each other, so it is non-reducing.
So galactose is the reducing sugar.