A Lost Logia is a piece of technology from an ancient world.
We have seen Lost Logia from Ancient Belka (Book of Darkness, Bible of Silver Cross) and Al Hazard (Jewel Seed, Relic, Saints Cradle).
Nanoha lives in Mid-Childa, and for them Ancient Belka is a lost world. For the inhabitants of Ancient Belka during the Belkan War, Al Hazard was a lost world to them as well. Due to the Belkan War you can see a lot of influences from Al Hazard as they tried to harness the immeasurable power of the Lost Logia of Al Hazard.
On the collapse of Ancient Belka, they tried to rebuild their empire using the Al Hazard Lost Logia, the Saint's Cradle. That was the Saint King Unification War with the lovable Olivie Sägebrecht. So that is a bit of history of the Lost Logia, a technology which has been lost.
Remember the 'magic' that Nanoha uses is technology as well, except that it is not mass-based (like a Nuclear Bomb), but magic-based (an energy coming from living beings, apparently) and was chosen technology for the Time Space Administration due to it being clean and relatively safe.