All types of multi-part payments in Lightning Network explained

Lightning Network uses source-based routing. What that means is that the sender is responsible for finding a route to the payee. But because the sender doesn't know the balances of all channels along a possible route, the proposed route can fail for lack of liquidity. So the sender is forced to try routes until it stumbles upon one that has enough liquidity to relay the payment; it's a process of trial and error. To increase the likelihood of success the idea of splitting up a payments into smaller parts was floated early on in the development of Lightning Network. The idea was that multiple smaller payments were more likely to succeed than one big payment. But you still want all separate parts of such a payment to act as one single atomic payment, that either succeeds completely or doesn't succeed at all.