If something's aside, it's not in the middle of things — it's off to the side, either literally or figuratively, as with the papers and books you push aside on the couch, to make room to sit.
When you ignore your conscience, you're pushing your best instincts aside. You can put money aside, to save it for the future. In a play, when an actor steps out of character and speaks directly to the audience, that's an aside. The expression "all joking aside," means "stop kidding around and get serious" as in, "All joking aside, you're not really going bungee jumping, are you?"