Structures that are the result of convergent evolution are called analogous structures or homoplasies; they should be contrasted with homologous structures which have a common origin.
An example of convergent evolution are the wings of insects, birds and bats. They all serve the same function and are similar in structure, but evolved independently. Eyes also evolved independently in various animals.
Another example are the aerial rootlets found in English ivy (Hedera helix) and wintercreeper (Euonymus fortunei) (and other vines). These rootlets are not derived from a common ancestor, but developed independently as an effective way to cling to whatever support the vine is climbing on.
Another example is the streamlined, fish-like shape of small whales. Except for the tail fins, these animals greatly resemble fish in outline, but are instead descended from four-legged land mammals. Their closest land relative today is thought to be the hippopotamus. Their modern shape is due to their water-based life cycle, as is the shape of the fish.
Insect mimicry is also an example of convergent evolution, as for example when an edible (palatable) butterfly develops a color pattern similar to a relatively unrelated inedible (unpalatable) butterfly, and by so doing escapes being eaten.