Terminology-history of p-adic representations Where appears for the first time the term Hodge-Tate representation.
Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.
 A: The notion of Hodge-Tate decomposition has been introduced by Tate, in 1967. 
(The paper itself is called $p$-divisible groups, and it appeared
in the Proceedings of a conference  on local fields that took place in Driebergen.)
There, he shows that over a $p$-adic field, the $p$-adic Tate module $T_p(G)$ of an Abelian variety with good reduction $G$ possesses a kind of Hodge decomposition:
$$ T_p(G) \otimes_{\mathbf Z_p} {\mathbf C_p} \simeq \mathbf C_p^g \oplus \mathbf C_p(-1)^g,$$
as Galois modules, $\mathbf C_p(-1)$ denoting the action via the cyclotomic character,
and $g$ being the dimension of $G$.
This is reminiscent of the Hodge decomposition over the complex numbers.
It has been proved later, by Faltings, that the $p$-adic étale cohomology of any 
smooth projective variety over a $p$-adic local field admits a similar decomposition
when tensored with $\mathbf C_p$:
$$ H^n(X,\mathbf Z_p)\otimes\mathbf C_p \simeq \bigoplus \mathbf C_p(i)^{h^{i,n-i}},$$
where $h^{i,n-i}=\dim H^i(X,\Omega_X^{n-i})$ are the Hodge numbers.
Now, there are other cohomology theories, the crystalline, the De Rham, etc.
and the rings forged by Fontaine play the rôle that $\mathbf C_p$ (technically,
the direct sum of all $\mathbf C_p(i)$) plays for the Hodge cohomology.
