Usually refers to New World Capiscum plants of the Solanaceae (nightshade) family, but can also refer to plants such as black pepper (Piper) and pimenta.
The word "pepper" refers to a range of unrelated plants, including Piper (black pepper) and pimenta, but it is usually used to refer to Capsicum of the Solanaceae family.
Capsicum include approximately 5-6 wild species, most of which have cultivars that are grown for their fruit. However, most chili pepper cultivars are C.Annuum cultivars - this includes most of the sweet peppers and a lot of the spicy varieties such as jalapeno and serrano.
Here is an example plant with many fruit: (C.Annuum cultivar 'Gypsy' - a heat tolerant small yellow bell pepper)