The similarity between an organelle and an organ lies in that their function within the bodies in which they operate is very specific. In the eukaryote cell (the type of cell that is one of the three branches of living beings), the organelles are different sub-parts within it which carry a specific function to protect it. Each organelle has an assigned duty within the cell. It can be anything, from producing the cell's energy (mitochondria), to conducting the process of homeostasis (vacuole). Also, each organelle works separately for the same purpose: To maintain the cell alive and functioning well.
Given that each organelle within the eukaryote cell works for the optimal function and welfare of the cell, we can compare it to the organs within the human (or animal) body. The human organs also work, both separately and together, using their own protective ways to remain healthy. Also in a similar way to the organelles, animal organs have specific functions. In humans and animals, for example, the heart pumps the blood, carrying air to the lungs. The lungs, on their part, do the function of providing oxygen to the entire body and to convert oxygen into the carbon monoxide that we breathe out after excreting it from the air we breathed in. The liver helps eliminate toxins, and each intestine is in charge of absorbing nutrients and eliminating waste.
In smaller words, organelles are the cell's "organs", and organs are the animal's version of organelles- to put it in a more colloquial and mundane way. :)