Where appears for the first time the term HodgeTate representation. Can i find somewhere explanation of the terminology HodgeTate, Derham etc. for representations and Fontaine's rings.

The notion of HodgeTate 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 $\mathbf C_p$: $$ H^n(X,\mathbf Z_p)\otimes\mathbf C_p \simeq \bigoplus \mathbf C_p(i)^{h^{i,ni}},$$ where $h^{i,ni}=\dim H^i(X,\Omega_X^{ni})$ 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. 

