Is a peanut technically a nut? If not, when people say they are allergic to "nuts", are they also allergic to real nuts,(walnuts, almonds, cashews, etc.)?
No, peanuts are not nuts in the botanical sense. They are legumes, much like peas or beans.
Chestnuts and acorns are examples of true nuts. Most of what we commonly refer to as nuts are botanically drupes, including walnuts, almonds and cherries, as well as some larger fruits like peaches (which are typically eaten for their flesh, rather than their seed).
To be clear, people sometimes use the term "tree nut" to distinguish actual botanical nuts and drupes from legumes; or "culinary nut" to include all of the nut-like foods.
While technically, peanut and tree nut allergies are different things, there is evidently a large correlation where people with one often have the other, according to Wise Geek.