The Warlock Spell 'Hex' (PHB 251) gives Disadvantage on ability checks.
Does this include saving throws and attacks? That is, in this context, are saving throws and attacks 'ability checks'?
Disadvantage from the Hex spell1 applies only to ability checks. The Player's Basic Rules and Player's Handbook differentiate ability checks from saving throws and attack rolls from ability checks in Chapter 7: Using Ability Scores. Specifically,
The three main rolls of the game—the ability check, the saving throw, and the attack roll—rely on the six ability scores. (PBR, p. 57; PHB, p. 173)
Under the subsection Advantage and Disadvantage, it again indicates that these are separate rolls.
Sometimes a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll. [Emphasis added] (Ibid.)
So while saving throws and attack rolls are based on your ability scores, they are not ability checks. In fact, Chapter 7 is dedicated to ability checks and saving throws (not attack rolls).
According PHB 173, ability check, saving throw and attack are threatened different:
Sometimes a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll.
So, the Hex Spell only cause disadvantage in ability checks (PHB 251).
Also, choose one ability when you cast the spell. The target has disadvantage on ability checks made with the chosen ability.