Place the egg, lemon juice, garlic, mustard and seasonings in a container and blend to combine (I like to use a stick blender).
With the blender running on low speed, pour the oil in slowly, allowing each addition to incorporate into the egg mixture before adding more. As more oil is incorporated, you can add the oil more quickly, working up to a slow stream.
As you add the oil, the mixture will thicken, changing from a liquidy mix of the beaten egg to a creamy-looking spread. If it's too thick, you can thin it a bit with some more lemon juice or warm water. Don't try to thin it with more oil - this will actually make it thicker.
Adjust seasonings to taste. Serve immediately or cover and chill for later use.
Here is a video of how I make aioli with a stick blender, making about double the quantity of this recipe.