Kali is worshipped in ordinary temple fashion by very large numbers of people who have nothing to do with Tantra, and she is also central to Tantric practice, and the two should not be collapsed.
Tantric traditions are commonly sorted into dakshinachara, the right-hand path, and vamachara, the left-hand path. The right-hand path uses conventional offerings and is continuous with mainstream temple worship. The left-hand path uses the five substances known as the panchamakara, whose names begin with the syllable ma, including wine, meat, fish and ritual sexual union, and it locates practice in places, times and materials that Brahmanical purity rules exclude. The doctrinal claim is that what is forbidden is not thereby powerless, and that a practitioner who has genuinely passed beyond attraction and revulsion is not bound by either. Whether the substances are used literally or substituted symbolically differs by lineage, and lineages disagree sharply about it.
Kali is also the first of the Mahavidyas, the ten great wisdom goddesses, a group including Tara, Chhinnamasta, who holds her own severed head and drinks her own blood, and Dhumavati, a widow, alone, inauspicious, riding a crow. The set is deliberately assembled from figures that a respectable pantheon would not include, and that is the point of it.
Two cautions. First, this material is initiatory: it is transmitted by a guru to a particular student, much of it is not published, and popular English accounts of it are frequently unreliable, drawing on nineteenth-century sources whose informants were telling outsiders what outsiders wanted to hear. Second, the modern Western use of the word tantra to mean a school of sexual technique bears almost no relation to any of this. Alexis Sanderson, David Gordon White and Hugh Urban have written on how that transformation happened.