Homology vs. Analogy

Homology
A homologous trait is when organisms share a trait due to common ancestry.

a. The bones of a human's front limb are homologous to the bones of a bat's front limb. 

b. Both limbs end in five digits, and even if some parts differ in size, both limbs have the same overall structure. However, they are used for different functions. It's obvious that humans cannot fly, but they use their five fingers to pick things up. Bats on the other hand, can fly but cannot pick things up with their five "fingers". 

c. All vertebrate animals (frogs, birds, fish, people) have skeletons because the ancestor of the vertebrates had a skeleton and passed that trait on to all of its descendants

Analogy

Traits that are similar due to reasons other than relatedness are called analogous. Such similarities can result from random chance or because unrelated organisms have adapted to similar environments. 

a. Although flying animals all have wings, if one would actually compare their wings, they'll find more differences than similarities. Insects have two pairs of wings, birds each have one pair. 

b. Insect wings lack bones, but a bird's wings, like an owl's, do have them. Butterfly wings are covered in scales, and bird wings in feathers. Although, they share the same function: they help the being fly. 

c. These organisms have adapted to life in the air and in doing so have evolved wings. Since wings have evolved independently in each of these groups, and don't indicate that they are closely related, the possession of wings is an analogous trait.