What specific rules elements are ability checks?
I know that skill checks are ability checks. Are saving throws considered ability checks?
No saving throws are not ability checks.
This almost sounds like you are asking the wrong question. In other words, you probably really want to know if 'power X' or 'spell Y' works in a certain situation, and because the power or spell says it modifies an ability check, you are now asking what counts as an ability check. You might get better results if you actually provide the context.
This is the one rules question that I have actually found myself having to answer at my gaming table, and have had to answer it more than once.
I have come to the simple answer that there are only three types of rolls that are called for in order to determine whether a taken action is successful or not: Attack Rolls, Saving Throws, and Ability Checks - none of which overlap with each other.
Undrhil wrote:
Well, I know Initiative is one.
I was more concerned about saving throws, because if I throw Hex on a creature, I can give it disadvantage on ability checks for one ability and then I could focus my other spells around the target making a saving throw with that ability, if saving throws counted as ability checks. Since it appears that they do not, this is all moot.
Thanks though. 
Undrhil wrote:
Undrhil wrote: