If something is ever-present it's always there, because it's continuous or habitual, like your dog's ever-present goofy grin and wagging tail.
If you spend winter in Newfoundland, Canada, you will experience the ever-present possibility of falling snow. In other words, you'll wake up each day to find that it's either snowing, or that it will probably start snowing soon. You could also describe your uncle and his ever-present flip-flops or your cousin and her ever-present Rubik's Cube. Anything that's constantly there is ever-present.