They don't do much, mostly just cosmetic. As you level you will randomly loot items with light/dark requirements, but they are not really better statted than other items you can get with no alignment. When the game launched they tried to make it that light/dark restricted what items visually a person could have, like certain colors of lightsaber crystals and such were restricted to specific alignment. Like most red sabers required dark. But people whined so BW caved and removed these restrictions on visuals.
The only unique thing for dark/light side are the relics and stuff like the exclusive mount/pet. Once you reach max level, the relics don't matter anymore as the upgrades are for anyone at that point. The main use for alignment really is for roleplay like purposes and for relics while you level. If you don't have an alignment of light or dark, you generally get no relics while you level. It's a large reason why players who are neutral want a neutral alignment added, as they are at a slight disadvantage while leveling through their class story. But really don't worry too much about it unless you really want too be strict on how you play or want those cosmetic items.
If you max out light or dark side on a character, it unlocks a new ability for your legacy that all your characters gain access too. For light side you get a buff that you can activate that gives you and your companion a 15% bonus to defense from what I recall. While the dark side legacy ability lets you hurt your companion to heal your character. These are nice little bonus that all characters get on your legacy even if they don't have the proper alignment, or even if the character who unlocked it changes their alignment.
