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[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (Berger). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (as discussed in the comments below, Berger reduced this to a result in the theory of $p$-adic differential equations, which was then proved independently by Andre, Kedlaya, and Mebkhout). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (Colmez--Fontaine). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (Berger). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

[Added: Just to set the scene, this question and answer are discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (Colmez--Fontaine). Also, unramified is a very strong condition in this setting, which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (Colmez--Fontaine). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.