What is a 'Key' in computer science?
In the most general sense, a key is a piece of information required to retrieve some data. However, this meaning plays out differently depending on exactly what situation you're dealing with.

In the contexts you mention, a key is a unique identifier for the complete data used to retrieve it from some location in the structure. Each key is associated with only one item, so it can be used to find a particular set of data. The data structure will usually be organized in such a way that finding the key is much more efficient than a linear search through all of the data. Sometimes the key is actually part of the data and stored along with it (like primary keys in the database); other times, it is segregated from the data itself (like in a hash map). The data structure will also often perform extra processing on the key (and only the key) to support its efficient searching algorithm (such as in a hash map, the key is converted into a hash code, or a database will index the primary keys using a B-tree).

In cryptography, a key is used in a sense more akin to physical keys used on locks. They're pieces of data required to obtain the original from the encrypted data (to "unlock" the data, so to speak).