A Warp Wire, also known as Ariadne Thread, is by far the most important item in the game. It is a normal-looking thread that you can buy at a low price on your local shop. However, if you are planning explore uncharted or dangerous areas, you absolutely need to have at least one, as they will warp you back to town no matter how far within the labyrinth you are. Only qualified explorers are allowed to buy these, possibly to keep greenhorn explorers close to town.
They are especially useful for escaping pinches related to FOEs, as they will instantly return you to safety even if you were cornered by one or more of them. They can't be used once you enter the fight, though.
These threads are consumed after a single use. One common and deadly mistake is to warp back to town but forget to replace the used thread, only to remember when you need them the most (most likely from chasing FOEs or critical condition).
Another related hazard can be found in Etrian Odyssey II in the form of cute squirrels living all around Yggdrasil. They are not hostile, but for some reason like threads a lot and will swiftly steal yours if you give them the chance, so NEVER PET THE SQUIRRELS. Otherwise you might find yourself stranded and return the hard way. Also, sometimes there is a shady guard not too far from the squirrels ready to sell you a warp wire... at 1000en each!
In Etrian Odyssey IV, there also is a guard in Lush Woodlands B1F, the first Stratum, who is willing to sell you an Ariadne Thread, but for 250en, which is more expensive than the shop's price, so only use it if you have ran out of threads, and are in dire need of one. However, if you haggle with him two or three times in a row, he will drop the price to 25en for the Thread, and 100en for the sword. Which makes both items less than in the store. It's worth noting that you can only buy one item from him before he leaves, not returning to the spot, so once you buy from him, don't think you can rely on him being there. There is also a similar item called the Silver Whistle that warps you to the entrance of that dungeon, but not back to Tharsis.
Alchemists and farmers can learn skills that work similar to a warp wire, providing you have TPs to spare; you still should carry one as insurance though, just in case something happens to the skill users, since their defense is low and they might get wiped out and leave you stranded.