You'll find all kinds of different styles of rum under the Gold Rum or 'golden rum' umbrella and what they all have in common is a lovely golden hue. Gold rum is also known as 'amber rum' too. This isn't a protected category or labelling method, nonetheless historically, colour has often been used as a way to categorise different rums. Distilled spirit is always colourless, so it's after distillation that the rum gains that golden hue. It could come from oak casks (usually bourbon) the rum was aged in. It could come from additional colouring like caramel. It could be both.
Generally, when people talk about golden rum, they're referring to medium-bodied, mellow expressions with delicate sweetness, but you'll find plenty of variation within the category.