partial derivative
Suppose you are standing on a hillside and you decide to walk due east only, never turning. The slope you feel under your feet — the rate at which your altitude changes per step east — is exactly a partial derivative. You freeze every coordinate except one, and measure the ordinary one-variable rate of change in that one direction.
Formally, the partial derivative of f with respect to x_i at a point a is the limit of (f(a + h e_i) - f(a))/h as h -> 0, where e_i is the i-th coordinate direction. We write it as df/dx_i (partial), or f_{x_i}, or D_i f. Computing it is just single-variable differentiation while pretending the other variables are constants.
Partial derivatives are easy to compute but treacherous as a measure of differentiability. A function can possess all its partial derivatives at a point and still fail to be continuous there. Partials only probe the coordinate axes; they say nothing about behavior along slanted directions, which is why the stronger notion of total derivative is needed.
For f(x, y) = x^2 y + sin(y): df/dx = 2xy (treat y as constant) and df/dy = x^2 + cos(y) (treat x as constant). At (1, 0): df/dx = 0, df/dy = 1 + 1 = 2.
Differentiate once per variable, holding the others fixed.
Existence of all partials does NOT imply differentiability or even continuity. But if all the partials exist near a point and are continuous there, the function is differentiable there (it is then called continuously differentiable, or C^1). Continuity of partials is the practical sufficient condition.