Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology.
An integral quadratic form has integer coefficients, such as x2 + xy + y2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x,y) ∈ Z if x,y ∈ Λ.
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
Read more about this topic: Quadratic Form
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