The Pfaffian of a skew-symmetric matrix is a square root of its determinant.
Let $A = (A_{i,j})$ be a skew-symmetric $(2n \times 2n)$-matrix with entries in some field (or ring) $k$.
The Pfaffian $Pf(A) \in k$ is the element
where
$\sigma$ runs over all permutations of $2n$ elements;
$sgn(\sigma)$ is the signature of a permutation.
Let $\Lambda_{2n}$ be the Grassmann algebra on $2n$ generators $\{\theta_i\}$, which we think of as a vector $\vec \theta$
Then the Pfaffian $Pf(A)$ is the Berezinian integral
Compare this to the Berezinian integral representation of the determinant, which is
Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of $N$ spinless electrons
where $pfaff(M_{k l})$ denotes the Pfaffian of the matrix whose labels are $k,l$ and $q= 1/\nu$ is the filling fraction, which is an even integer. For Pfaffian state see
J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026
Claudiu Raicu, Jerzy Weyman, Local cohomology with support in ideals of symmetric minors and Pfaffians, arxiv/1509.03954
Haber, Notes on antisymmetric matrices and the pfaffian, pdf
There is also a deformed noncommutative version of Pfaffian related to quantum linear groups:
Pfaffian variety is subject of 4.4 in
Relation to $\tau$-functions is discussed in
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