Cracking the eggs into the skillet before it's hot enough
When frying an egg, the butter in the pan should be bubbly and foamy; if it's not, your pan isn't hot enough yet, and your eggs won't fry evenly.
Not covering the skillet
There's nothing worse than ending up with a perfectly runny yolk and viscous, undercooked whites. If you aren't planning on going the over-easy route, cover the skillet. This traps the heat around the egg, which ensures that the top of the whites firm up while the bottom remains crisp.
Pro tip: If butter isn't your jam, try frying eggs in extra-virgin olive oil; it's (somewhat) healthier and will create a crispy, lacy edge around the whites. Heat 2 tablespoons of the good stuff into a skillet over medium-high heat and let sit for 1 minute. Crack an egg into the center of the skillet; the oil should immediately start bubbling around the whites. Cook until the edges appear crisp and golden brown -- about 2 minutes. Season with fresh-cracked pepper and salt.