The hazelnut is the fruit of the hazel tree and therefore includes any of the nuts deriving from species of the genus Corylus, especially the nuts of the species Corylus avellana. They are also known as cobnuts or filberts according to species.
Ripe hazelnuts
Cracked hazelnut shell displaying the edible seed
Hazelnut tree, Turkey
Hand-harvest and sun-drying of hazelnuts in Turkey
A nut is a fruit consisting of a hard or tough nutshell protecting a kernel which is usually edible. In general usage and in a culinary sense, a wide variety of dry seeds are called nuts, but in a botanical context "nut" implies that the shell does not open to release the seed (indehiscent).
Chestnuts are both botanical and culinary nuts.
Nuts being sold in a market
Raw mixed nuts, sold as a snack food
Image: Almonds in shell, shell cracked open, shelled, blanched