The Lifeboat is a prototype/back up time machine created by Mason Industries and is also the one that Lucy Preston, Wyatt Logan, and Rufus Carlin use to chase Garcia Flynn. The Lifeboat was upgraded by Rufus and Jiya to be able to hold four passengers safely.
The Mothership is a more advanced time machine that was stolen by Garcia Flynn, Stiv and Anthony Bruhl. They used the Mothership to change key events in history to hurt/erase the Rittenhouse organization. After learning what Rittenhouse wanted to do with the time machines, Anthony wanted to destroy the Mothership and encouraged Lucy to do the same to the Lifeboat, but was shot and killed by Flynn when he attempted to. The new pilot of the Mothership is Emma Whitmore.
The Future Lifeboat is a time machine that appears to be the original Lifeboat, but upgraded. All of its upgrades are unknown, but what is known is its ability to travel to time periods where its passengers already exist.