A roll under mechanic is almost a mirror to the roll over mechanic. It is most commonly used against an ability of the character, in which case they must roll lower than their ability to pass.
Note that this doesn't always mean "the lower the roll, the better": Some games can rely on rolling as high as possible as long as it's under the limit (like in The Price is Right); this makes one's ability score an "upper bound" of how well they can succeed.
Games that use a Roll under mechanic
- Advanced Dungeons & Dragons (except for attack rolls)
- BESM (well, first two editions did), other tri-Stat system games
- Dark Heresy and all related systems.
- Eclipse Phase
- Warhammer 40000 for Leadership tests and attribute tests.
- WarmaHordes for the same, plus skill tests.
- The Dark Eye
- Call of Cthulhu
- Gamebooks, like the Fire*Wolf series.
- In Nomine Satanis / Magna Veritas (and possibly In Nomine too)
- Fading Suns (with TPiR mechanic)
- Pendragon (with TPiR mechanic)
- Star Trek Adventures (and all other games using Modiphius's 2d20 system)