Why is it that gravity does not pull a bird or aeroplane back to earth?

Gravity does affect birds and airplanes the same as it does to any object. But these objects are using their wings or engines to exert an stronger force than the one gravity is exerting on them.

g = 9.81 m/s2 = 32.2 ft/s2

This means that, ignoring air resistance, an object falling freely near the Earth's surface increases its velocity by 9.81 m/s (32.2 ft/s or 22 mph) for each second of its descent. Thus, an object starting from rest will attain a velocity of 9.81 m/s (32.2 ft/s) after one second, 19.6 m/s (64.4 ft/s) after two seconds, and so on, adding 9.81 m/s (32.2 ft/s) to each resulting velocity. Also, again ignoring air resistance, any and all objects, when dropped from the same height, will hit the ground at the same time.