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.

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Hodge-Tate appears (in that name) at least as early as Serre's book "Abelian $\ell$-adic representations" (1968). Someone with a better knowledge of the literature may be able to beat that. – tkr Oct 25 '11 at 21:45

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.