The term "teleporting trains" is conjectural. The real name is unknown, and a non-canonical nickname or title is used instead. See the reasons for this title in the "Behind the scenes" section below and/or the relevant discussion on the talk page, if there are any.
Allgon City's teleporting trains were a transit system of teleporter monorail like trains which teleport throughout the city. Ratchet had to ride the train rails to get to a piece of the Hypnomatic. It was a wild ride, as he had to jump from train to train and rail to rail. Whenever he had to jump on a train, he would have to be quick, as they warped very quickly. There was also an area in which Ratchet had to dodge a wave of speeding trains, by jumping from rail to rail back and fourth before continuing. Some of the lines were broken (probably due to the Protopet attack), so some of the trains went into the water.