Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say, ${\bf R}^n$, are (in principle, at least) generated by the position operators $x_j$ and the momentum operators $\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$ \frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where $[,]$ here is the commutator $[A,B] = AB-BA$.
Could it be explained what Tao means by linear partial differential operators are generated by the position and momentum operators?
Think of a polynomial in $2n$ variables $P(x_1,...,x_n,y_1,...,y_n)$. Since we can compose operators this corresponds to multiplication and we can think of evaluating the polynomial where $x_j$ are the position coordinates and set $y_j=\frac{\partial}{\partial x_j}$. This is related to the concept of the 'symbol' of an operator. There's also something you can say about such polynomials (I think you need to assume they do not involve $x_j$'s only $y_j$) and the Fourier transform. Such operators will be constant coefficient partial differential operators. When you take the Fourier transform of such an object, every derivative gets mapped to a polynomial in the Fourier variable say $\widehat{\left(\frac{\partial^n}{\partial x_j^n}\right)}= (2\pi i \xi_j)^n$. This establishes a correspondence between constant coefficient partial differential operators (as well as PDE) and polynomials. In this case, the polynomial the operator gets mapped to is usually called the symbol. Hope this helps.