A broadside is the long-side of a Ship. Therefore, the location for most of the ship's Firepower. The broadside has more armor than the Bow and Stern; however it's also the weakest side of a ship to Grapple each other.
If you are Linefighting, most of the time you are trying to broad-to-broad. This means you fire your broadside(s) against the enemy's broadside(s). With this tactic, the side with the best discipline (in terms of Concentrated Fire) and sheer Firepower would win. Making Juggernaut Ships extremely useful if you can bring your cannons to bear!
It's recommended to increase the value of your broadside with Fittings.
Also, "firing your broadside" means fire all cannons at that side.
This information is provided by the update that PotBS Wikia is undergoing, by Captain Vuur.