It is instructive to work up to this ``one dimension at a time.''
For simplicity we will stick to using 
as the symbol
for the function of which we are taking derivatives.
Let the dimension be x. Then we have no ``extra'' variables
to hold constant and the gradient of 
is nothing
but 
. We can illustrate the
``meaning''
of 
by an example: let 
be the mass of an
object times the acceleration of gravity times the height h of
a hill at horizontal position x. That is, 
is the
gravitational potential energy of the object when it is
at horizontal position x. Then
Note that 
is the slope of the
hill and
is the horizontal component of
the net force
(gravity plus the normal force from the hill's surface)
on the object. That is, 
is the
downhill force.
In the previous example we disregarded the fact that most hills
extend in two horizontal directions, say x = East
and y = North. [If we stick to small distances we won't notice
the curvature of the Earth's surface.]
In this case there are two components to the slope:
the Eastward slope 
and the Northward slope 
.
The former is a measure of how steep the hill will seem
if you head due East and the latter is a measure of how steep
it will seem if you head due North. If you put these together
to form a vector ``steepness'' (gradient)
then the vector 
points uphill
--- i.e.
in the direction of the steepest ascent.
Moreover, the gravitational potential energy 
as before [only now 
is a function of
2
variables, 
] so that 
is once again
the downhill force on the object.
If the potential 
is a function of
3 variables,
[such as the three spatial
coordinates
x, y and z --- in which case we can write it a little
more compactly as 
where 
, the vector distance
from the origin of our coordinate system to the point in space
where 
is being evaluated], then
it is a little more difficult to make up a ``hill'' analogy
--- try imagining a topographical map in the form of
a 3-dimensional hologram where instead of lines of
constant altitude the ``equipotentials'' are
surfaces of constant 
. (This
is just what
Physicists do picture!) Fortunately the math
extends easily to 3 dimensions (or any larger number,
if that has any meaning in the context we choose).
In general, any time there is a potential energy
function 
we can immediately write
down the
force 
associated with it as
A perfectly analogous expression holds for the
electric field 
[force per
unit charge]
in terms of the electrostatic potential 
[potential energy per unit charge]:
Although we won't be needing to go beyond 3 dimensions very often
in Physics, you might want to borrow this metaphor for application
in other realms of human endeavour where there are more than 3
variables of which your scalar field is a function.
You could have 
be a measure of
happiness,
for instance [though it is hard to take reliable measurements
on such a subjective quantity]; then 
might be a function
of lots of factors, such as 
= freedom
from violence,
= freedom from hunger, 
= freedom from poverty,
= freedom from oppression, and so
on.
Note that with an arbitrary number of variables we get away from
thinking up different names for each one and just call the
variable ``
.''
Then we can define the gradient in N dimensions as
