The decimal numbers we play with in grade school are called the real numbers and can be modeled as points on a single line we call the real number line. On the other hand, complex numbers are two-dimensional and are modeled as points on a 2D surface like a computer screen. Traditionally we talk about the left-to-right placement of the number as the real component and the bottom-to-top placement as the imaginary component. To add two complex numbers we just add their real components together and then their imaginary components together.
Drag the green and blue points and watch how they sum to the red point. The "choose" radio buttons let you pretend you're sliding the green vector (arrow) from the origin to the tip of the blue vector or the other way around. Either way, you end up at the red point: red = green + blue, or red = blue + green. Mathematicians call this phenomenon commutativity.
Snapshot 1: It's not wrong to think of addition of complex numbers as the sliding of the end of one vector to the tip of a second vector. Here we slide the green vector to the tip of the blue vector.
Snapshot 2: Here we slide the blue vector to the tip of the green vector. In both cases we end up at the red point, which demonstrates that addition in the complex number field is commutative.
Snapshot 3: The horizontal line in the middle of the graph is where all the numbers lie which have imaginary components equal to zero. These numbers are none other than the real numbers we first learn about in grade school. Adding and subtracting green and blue vectors that lie completely on this real number line results in a red vector which also lies on the real number line. Because of this self-contained quality mathematicians would say that the real numbers are closed under addition.