Homologous- same evolutionary origin and different functions.

Analogous- different evolutionary origin but same function.

Examples of homologous organs are:
1. Wing of a bat and our forearm, or any upper limb of any mammal with other mammals. These structures have similar arrangement of bones but perform different functions according to their habitat adaptation.
2. In plants, all the stem modifications or all the root modifications are homologous with each other. Such structures serve different functions like support, food storage et cetera.

Analogous organs have the following examples:
1. Leaf tendril and stem tendril. Both these perform the function of coiling or support. But one is of foliar origin and the other if stem.
2. Wings of a bat and that of an insect. Bat wing is a bony structure with association with underlying muscles. Insect wing are, however, membranous extensions.