You will face the Slug boss near the end of Chapter Eight.
When you get into the ADS Cannon, you'll see the large Slug Boss attached to the ship. Fire at it once to wake it up and begin the boss fight.
From his body the Slug extends five tentacles, the same as the ones used by the Leviathan boss. As with that boss fight, the objective is to shoot the large yellow bulge until it explodes and severs the tentacle.
To attack, the Slug reaches down with each tentacle and grabs a part of the ship (either a random metal piece, or something similar to a fire extinguisher), then lifts the tentacle up and throws the part at you. Similar to the previous ADS cannon battle, you'll be able to shoot the piece out of the air before it hits you and causes damage.
To prevent the tentacles from throwing the parts at you, you can either fire at the tentacle while it's digging for pieces or hit the tentacle after it's lifted up the piece and it's preparing to throw. This will either cause it to drop the piece, or screw up the throw so the piece floats off in a safe direction. You don't necessarily need to hit the yellow part to stun it, but too many shots to the non-yellow part will cause the Slug to retract that tentacle. Try to ignore any of the pieces that aren't flying in your general direction to conserve ammo and time.
Since the Slug is some distance from you, there is a slight time difference between you firing and the shot hitting, so if the tentacles move right after you fire, don't expect the shot to hit. Whenever you have a moment and one of the tentacles is standing still, unload with both barrels at the yellow bulge in an attempt to destroy it. Once a tentacle is destroyed, it does not come back, so once you've taken out the first two tentacles, the battle gets extremely easier, as it's easy to keep the Slug from picking up new parts to throw, and you can simply pick apart what's left.
Complete this fight will unlock the Get off my ship! achievement, and completing it with more than 50% shield remaining (this one starts at 100) will also unlock the Slugger achievement.