Main difference between Riemannian geometry and Pseudo-Riemannian geometry First of all I want to know: what is the main focus of each subject, what we study in each one?
And secondly, why and how they are different?
Thirdly, what's the connection between them?
The answers I am looking for are more intuitively and plain simple, rather rigorous.
 A: A Riemannian manifold is a manifold $M$ equipped with a Riemannian metric, that is, a field of positive-definite inner products on the tangent spaces. On the pseudo-Riemannian metric, one replaces "positive-definite" with "non-degenerate", meaning that $\langle v,w\rangle = 0$ for all $w$ implies $v=0$. In other words, we still require the identification between tangent spaces and their duals, given by the metric, to be an isomorphism (and positivity plays no role on this). In several coordinate computations, the metric is represented by a matrix $(g_{ij})$, and the inverse $(g^{ij})$ appears. We are, instead of requiring the matrix $(g_{ij})$ to be positive-definite, requiring it to be merely non-singular, so that $(g^{ij})$ will still exist, and we carry on.
One of the main consequences is that if $v$ is any tangent vector, we no longer have $\langle v,v\rangle \geq 0$, but $\langle v,v\rangle$ may be zero, or have any sign, even when $v$ is non-zero. Call $v$ spacelike when $\langle v,v\rangle$ is positive, lightlike if zero, timelike if negative.
In Physics, such metrics are used in relativity. Here's a quick intuition: assume that $v = (\Delta x,\Delta y, \Delta t)$ is the displacement vector from two points in the plane, each one carrying a clock, and the time displacement is $\Delta t\neq 0$. Let $c=1$ be the speed of light. Then since there cannot be any speed greater than the speed of light, we have $$\left(\frac{\Delta x}{\Delta t}\right)^2+\left(\frac{\Delta y}{\Delta t}\right)^2< 1,$$which becomes $$(\Delta x)^2 + (\Delta y)^2 - (\Delta t)^2 < 0.$$The left side is quadratic in the variables $\Delta x$, $\Delta y$ and $\Delta t$, so polarizing it leads us to consider the Lorentzian scalar product $$\langle v,w\rangle_L = v_1w_1+v_2w_2-v_3w_3$$in $\Bbb R^3$. Then $\Bbb R^3$ equipped with this product is called Minkowski space. Of course you can do this in dimensions $3$, $4$ or $n$, whatever. If $v$ is timelike, its orthogonal complement (the restspace of the observer, or simultaneity plane) has only spacelike vectors, and so Euclidean distances between points (events!) in such plane can be measured (according to $v$). And then people go and develop the theory from things like that.
