Airplane and aeroplane are both correct spellings depending on what type of English you are writing.
How a bird flies is quite complicated, but a bird's wing is much the same as an airplane's wing. The same principles of pressure apply. The difference is that a bird can flap it's wings to produce lift when standing still and an airplane must rely on airspeed to generate lift.
When a bird flaps its wings, it doesn't just move them straight up and down. In addition to moving the whole wing up or down, the bird will also twist the wing, and "fold" the wing. Folding means decreasing the wingspan.
When a bird flaps down, it also twists the leading edge of the wing down. This has the effect of generating airflow over the wing from the leading edge to the trailing edge. The twisted wing also orients the lift vector forward, providing thrust for the bird.
When a bird flaps up, it twists the leading edge of the wing up, again generating airflow from the leading to trailing edge. However, because the wing is twisted up, the lift vector is now pointing behind the bird, adding drag and slowing the bird down. If the bird is not landing, this is undesirable, so the bird will "fold" it's wings at the same time, making the wingspan less and the generated drag less. Conversly, when landing or slowing down, the wings would be folded on the down stroke, generating less thrust.