For a function f:U→R (U⊆Rn some open set) and a point x=(x1,...,xn), define

fi,x(t) = f(x1,...,xi-1,t,xi+1,...,xn).
Then fi,x is a function of one variable when t is near xi, and we can do calculus with it. The partial derivative of f with respect to xi is defined to be
f/∂xi = dfi,x/dt.
Note the unfortunate double usage of x and especially xi to denote both a vector or a scalar variable and a particular value of that variable. Unfortunately, this practice is so common that I feel I must abide by it. This is a function ∂f/∂xi : U→R. If the derivative on the RHS does not exist, the partial derivative on the LHS doesn't, either. By tradition, the same letters are re-used, instead of new ones like t. Your multivariable calculus textbook may well follow this "convention", and give a seemingly different definition. The only purpose of this is to make a confusing subject more confusing.

Another notation for ∂f/∂xi is fxi. This notation is particularly useful for taking multiple partial derivatives: for a function f(x,y),

fxx = ∂2f/∂x2 = ∂/∂x(∂f/∂x);
fxy = ∂2f/∂y∂x = ∂/∂y(∂f/∂x);
fyx = ∂2f/∂x∂y = ∂/∂x(∂f/∂y);
fyy = ∂2f/∂y2 = ∂/∂y(∂f/∂y).
In particular, the Laplacian of f is Δf = fxx+fyy.

WHEN f is differentiable, it turns out that

f = (∂f/∂x1,...,∂f/∂xn).
But even if all partial derivatives exist at x, f might not be differentiable there -- be careful!

Contrary to what you might expect (or even read about in various places, once upon a time even on E2...), it is not necessarily true that ∂2f/∂xi∂xj = ∂2f/∂xj∂xi. You cannot, in general, interchange the order of the derivatives.