Probably the best place to start is to check the handbook (something that I forget to do all the time). The DnD 4e Player's Handbook actually gives a decent description and several examples of the common actions of each skill.
But abstractly, you can't really set aside the ability scores that the skills are based on. What the skill is and how it's performed is informed by the ability score. Personally, when I'm trying to decide which would cover an action I think this: does this action require muscle, or precision?
Strength is all about muscle, and athletic people are going to be strong from their exercise. To jump really high an athletic person needs large leg muscles, to break ropes tying her down an athletic person needs large arm muscles. But neither of them are exactly delicate maneuvers.
Dexterity is about agility and practice, and acrobatic people are going to be precise. To slide through an enemy's legs without getting hurt requires precision, as does walking a tightrope. But you don't really have to be strong to do either.
Some things honestly require both: Someone trying to jump to a really high window AND through it without touching the broken glass on the sides should probably roll both, and failure consequences probably depend on which roll was failed.
I think the confusion comes from real life examples; most people that we think of as acrobatic (gymnasts, martial artists, and uh, acrobats) are also athletic because they do things like jumping, climbing and running, but that doesn't mean that the athletic actions are also acrobatic.