The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity

To remind it, it requires some notation. Let $x_{ij}$, $1\leq i,j\leq n$, be commuting variables. Define differential operators on the space of functions on $n\times n$ matrices: $$E_{ij}=\sum_{a=1}^n x_{ia}\frac{\partial}{\partial x_{aj}},\, 1\leq i,j\leq n.$$

The Capelli identity states that $$\det\left[\begin{array}{cccc} E_{11}+n-1&\dots&E_{1,n-1}&E_{1n}\\ \vdots&\vdots&\dots&\vdots\\ E_{n-1,1}&\dots&E_{n-1,n-1}+1&E_{n-1,n}\\ E_{n,1}&\dots&E_{n,n-1}&E_{n,n}+0 \end{array}\right]=\\\det\left[\begin{array}{ccc} x_{11}&\dots&x_{1n}\\ \dots&\dots&\dots\\ x_{n1}&\dots&x_{nn} \end{array} \right]\cdot \det\left[\begin{array}{ccc} \frac{\partial}{\partial x_{11}}&\dots&\frac{\partial}{\partial x_{1n}}\\ \dots&\dots&\dots&\\ \frac{\partial}{\partial x_{n1}}&\dots&\frac{\partial}{\partial x_{nn}} \end{array}\right]. $$

Note that in the right hand side of the equality the two matrices have commuting entries, while in the left hand side the entries do not commute. Hence one has to be careful to define the determinant. **The convention for the determinant of such matrices is
$$\det(a_{ij})=\sum_{\sigma\in S_n}sgn(\sigma) a_{1\sigma(1)}a_{2\sigma(2)}\dots a_{n\sigma(n)},$$
where the order of terms is important.**

**This order of terms in the determinant is the most mysterious point for me. Is there any reason for it? What happens if one chooses some other ordering of terms: will the Capelli identity be modified somehow or it will not work at all?**

There are more recent generalizations of the Capelli identity, see e.g. the above link to Wikipedia. However unfortunately they do not clarify to me the original Capelli idenity, but rather use it as a basic inspiration for further extensions (may be I am missing something).

**UPDATE.** So far the most conceptual approach to the Capelli identity I was able to find in the literature is due to I. Gelfand and V. Retakh, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 13--25 (Section 3.4). This is their first paper on their general theory of non-commutative determinants. They use it to rewrite the Capelli identity in their language. I have not studied their proof in detail, but apparently they do not use this strange and seemingly arbitrary definition of det as a sum over all permutation of terms with prescribed ordering. I still have to understood how difficult and natural it is to pass from their language to the classical one.