What is the difinition of $f:X\to Y$ a fiberation? Is there a common setting for $X,Y,f$?
In some book fiberation is defined (1)for varieties $X, Y$ projective and smooth, each fiber is connected and $f$ surjective. Some required (2)$X$ integral, $f$ proper or projective and flat, if $Y$ is a curve it is dedekind.
Does (1) implies $f_*(O_X)=O_Y$ or flatness? Does $f_*(O_X)=O_Y$ implies flatness? Does (2) implies fiber conectness? Is surjective needed in (2) or implied? If the map is surjective, then is the map called faithfully flat?
And thanks for references on fibrations!
Fibration is not (in my experience) a precisely defined term in algebraic geometry; maybe particular authors give precise definitions in various contexts though.
To answer some of your other questions:
smooth implies flat, so the condition with smoothness is more stringent than the condition with flatness.
faithfully flat means flat and surjective.
If $f: X \to Y$ is proper and surjective with $X$ and $Y$ integral and $Y$ normal (e.g. smooth), and with its generic fibre both geometrically connected and geometrically reduced, then $f_* \mathcal O_X = \mathcal O_Y$. (By properness, $f_*\mathcal O_X$ is a sheaf of coherent $\mathcal O_Y$-algebra. Since $f$ is surjective, it is torsion-free over $\mathcal O_Y$. By the assumption on the generic fibre, we see that $f_*\mathcal O_X$ has rank one over $\mathcal O_Y$, and so by normality equals $Y$. Note that that geometric reducedness is important; in characteristic $p$ Frobenius maps give examples of finite (in particular proper) maps with connected fibres where $f_*\mathcal O_X$ is not equal to $\mathcal O_Y$. In char. zero, the geometric reducedness of the generic fibre is automatic, given the assumption that $X$ is integral.)
Note that $f_*\mathcal O_X = \mathcal O_Y$ implies connected fibres when $f$ is proper, by the theorem on formal functions.
$f_*\mathcal O_X = \mathcal O_Y$ is not particular related to flatness one way or another. E.g. it holds when $Y$ is normal and $f$ is birational (by the third point, since birational means that $f$ is generically an isomorphism), but birational morphisms are not flat if they are not the identity (since the fibre dimension jumps from $0$ at most points to $> 0$ at certain points). On the other hand, any map of smooth varieties with $0$ dimensional fibres is flat (a special case of so-called miracle flatness), but the fibres are typically not connected; think about morphisms between smooth curves.
Note that a proper flat morphism will be surjective onto a union of connected components of $Y$ (and in particular, will be surjective if $Y$ is connected). The reason is that proper morphisms are closed (by definition), and flat morphisms are open (under mild finiteness conditions, e.g. locally of finite presentation, so certainly for morphisms of varieties); so the image of a proper flat morphism is both open and closed.