When certain spells or other game effects request skill-less checks
As pointed out by Someone_Evil, certain spells, such as entangle, dispel magic, telekinesis, etc, require ability checks to use or escape them, and these ability checks do not specify a skill.
For each spell of 4th level or higher on the target, make an ability check using your spellcasting ability. The DC equals 10 + the spell's level. On a successful check, the spell ends.
A creature restrained by the plants can use its action to make a Strength check against your spell save DC. On a success, it frees itself.
Make an ability check with your spellcasting ability contested by the creature's Strength check. If you win the contest, you can move the creature...
When rolling Initiative to determine your order in combat
There's also Initiative rolls (thanks Greenstone Walker), which is technically a Dexterity ability check, although because it has it's own name ("Initiative"), it's often not called that. Initiative is described in the basic rules under the Order of Combat section:
When combat starts, every participant makes a Dexterity check to determine their place in the initiative order.
Although you cannot add your proficiency bonus to it, there are other bonuses you can add to it.
When you make an ability check using a tool proficiency
As pointed out by V2Blast, "ability checks relating to a tool do not have any skill associated with them, because whether you add your proficiency bonus to the roll simply depends on whether you're proficient with the given tool. The most obvious example in the rules is thieves' tools, which you can use to make a Dexterity check to pick a lock/disarm a trap; that's not associated with any skill, but you can still add your proficiency bonus depending on whether you're proficient with the tool."
From the basic rules on Tools:
Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.
In other words, using a tool you're proficient with is just an alternative way to add your proficiency bonus vs. making a skill check. Either way, it will be a kind of ability check that uses one of your six ability scores (but not necessarily always the same one, as mentioned in the above quote) and optionally your proficiency bonus if you're using a skill or tool set that you're proficient with.
When the DM otherwise decides that none of the existing skill or tool proficiencies apply to the particular check
Without wanting this to turn into a list of every example I can think of, basically, when there are situations that call for an ability check, using one of the 6 ability scores, but the nature of the check doesn't really relate to a specific skill (or tool set, as per the above section).
In such scenarios the DM would ask for an "Intelligence check", for example, rather than, say, "Intelligence (History)" (or more commonly just "History"). In practice, associating an ability check with a skill just allows the player to be able to add their proficiency bonus if they are proficient in that skill, so an ability check without an associated skill (or tool) simply wouldn't include your proficiency bonus.
As an example that I've seen other DMs use, a DM might ask for an Intelligence check to see if your character can remember something that happened last week. Of the five skills normally associated with Intelligence (Arcana, History, Investigation, Nature, Religion), none of those really apply (History is the closest, but that's more to do with having studied history, not simply remembering something from the recent past).
Another example that I've seen is whether a not very intelligent character can think of a plan that the player (who is more intelligent that their INT 8 character) has thought of, to see if that character was able to have the brainwave necessary to think of it. These sorts of ability checks can literally be as arbitrary as that, depending on the DM.