Fruits that people commonly mistake for vegetables: Tomato/catsup, cucumber/pickle, pepper, gourds (squash, pumpkin, etc.), grain (corn, etc.), avocado, etc.

A fruit is the part of the plant that contains the seeds (or are the seeds themselves, such as with grain) and vegetables are everything else.

Even more confusing are fruits like strawberries that have the seeds seemingly on the outside. In fact, it is a cluster of fruits fused together, like pineapples. Also, bananas (which I believe are sterile) are fruits despite not containing seeds. There is some technical definition based on plant anatomy involving words like pericarp and oviduct (joke:).

According to GeneralWesc's definition, peas, green beans, lentils, and chile are also fruit. However, I'm not so sure that peas, lentils and corn are really fruit per se, or if they're nuts. Or perhaps nuts are a subcategory of fruit. More when I find the answer.

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Contrary to popular belief, bananas do contain seeds. If you examine the last quarter of a banana (away from the end you peel) you will notice tiny black dots arranged in a ring. These, as far as I can tell, are banana seeds.