What does the symbol nabla indicate?

We may think of $ \nabla $ as an operator ( del operator ) in the following sense.

It takes a function $f$ and turns it into a vector $\nabla f$ .

$\nabla f= \left\langle \frac {\partial f}{\partial x},\frac {\partial f}{\partial y}, \frac {\partial f}{\partial z} \right\rangle $ is called the gradient vector.

The gradient vector points to the direction at which your function increases most rapidly.

For example if $$ f(x,y,z)= x+3y^2 -10z$$ Then $$ \nabla f (x,y,z)= \langle 1,6y,-10\rangle $$

and if there is a point given, say $(1,3,5)$, we can evaluate $ \nabla f (1,3,5)= \langle 1,18,-10\rangle.$

This vector points at the direction of maximum increase of our function at $(1,3,5).$

Nabla is a vector whose components are operators. In the three-dimensional case you quote, $\nabla=(\partial_x,\partial_y,\partial_z)$. It is not a vector in the usual sense (of vectors in $\mathbb R^3$), but it is a very convenient abuse of notation.

The example given in the question gives a convenient way to write the gradient of a function $f:\mathbb R^3\to\mathbb R$ as $$ \nabla f(x,y,z) = (\partial_xf(x,y,z),\partial_yf(x,y,z),\partial_zf(x,y,z)). $$ As it turns out, this kind of a derivative is useful.

If you have a function $g:\mathbb R^3\to\mathbb R^3$, there are two typical derivatives you will need. One of them is the divergence, which is the scalar quantity $\partial_xg_x(x,y,z)+\partial_yg(x,y,z)+\partial_zg(x,y,z))$. It is convenient to write this as $$ \nabla\cdot g(x,y,z), $$ since the formula does indeed look like an inner product of the vector $g=(g_x,g_y,g_z)$ and our $\nabla$.

The other one is the curl, which is given in terms of components as $$ (\partial_yg_z-\partial_zg_y,\partial_zg_x-\partial_xg_z,\partial_xg_y-\partial_yg_x). $$ (I omit the arguments for brevity.) This one looks like a cross product, and it is indeed typical to write it as $\nabla\times g(x,y,z)$.

The point is that there are these three basic instances where it is convenient to think of $\nabla$ as a vector of operators, even if such objects aren't studied in general.

How you've described it, it's used as the gradient of a function in multivariable calculus.

By itself, the nabla can be thought of as a vector of partial derivative operators, and when applied to a multivariable function, it represents the vector of partial derivatives of each component (dot product) and the direction of steepest ascent for some input:

$$\nabla = \begin{bmatrix}\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \\ \vdots\end{bmatrix}$$

$$\nabla f(x, y, z,\dots) = \begin{bmatrix}\frac{\partial}{\partial x}\,f(x, y, z,\dots) \\ \frac{\partial}{\partial y}\,f(x, y, z,\dots) \\ \frac{\partial}{\partial z}\,f(x, y, z,\dots) \\ \vdots\end{bmatrix}$$

The nabla can be applied to a number of different areas in multivariable calculus, such as divergence or curl. In all these cases, the nabla can be treated like a vector which you can dot or cross with another vector, such as a multivariable function. That said, it is an operator.