What are Merklized Alternative Script Trees? I have seen the acronym MAST floating around. I saw that it stands for Merklized Alternative Script Trees, but what does that mean? In what context do they appear and what can they be used for?
The idea of Merklized Alternative Script Trees (or previously Merklized Abstract Syntax Trees) dates back to an idea Russell O'Connor had in 2012. The original idea was that you could have alternative scripts or script fragments stored as leaves in a Merkle tree and then the leaves of the tree that weren't used could be pruned away and not take up valuable space on the blockchain. Compare this to P2SH (Pay to Script Hash) where the entire script is hashed and then revealed on the blockchain at spend time. This brings block space efficiencies (and hence lower fees) as well as privacy benefits.
The earlier formalized proposals for MAST were BIP 114 and BIP 116 and the most recent proposal is of course Taproot (BIP 341). The reason why we moved away from using Merklized Abstract Syntax Tree terminology is that with Taproot (and BIP 114) only one single leaf of the Merkle tree can be executed. You cannot split one particularly long script path into multiple script fragments stored on different leaves of the tree and then satisfy a combination of these leaves. It is effectively a tree of ORs (no ANDs of different leaves and no IF, ELSEs that lead to satisfying different leaves on the tree). You can have ANDs and IF ELSEs but they are contained within leaves not between leaves. Satisfying a single leaf is sufficient to spend the UTXO. As Johnson Lau says in BIP 114:

Only one branch is allowed for execution, and users are required to transform a complicated condition into several mutually exclusive branches.

Pieter Wuille and Russell O'Connor discussed why this design decision was made for Taproot at London BitDevs in July 2020. One of the reasons was that you only need a logarithmic number of Merkle branch nodes to get to any given leaf. There is a combinatorial explosion if there is the potential to have different combinations of all the leaves. Another reason is that the design space had to be limited at some stage to give the proposal the best chance of obtaining community consensus and ultimately be activated.