EDITS MADE BASED ON FEEDBACK
A key is unique to a row. It's a way of identifying a row. Rows may have none, one, or several keys. These keys may consist of one or more columns.
Keys are indexes with a unique constraint. This differentiates them from non-key indexes.
Any index with multi-columns is called a "composite index".
Traditionally, a primary key is viewed as the main key that uniquely identifies a row. There may only be one of these.
Depending on the table's design, one may have no primary key.
A primary key is just that - a "prime key". It's the main one that specifies the unique identity of a row. Depending on a table's design, this can be a misnomer and multiple keys express the uniqueness.
In SQL Server, a primary key may be clustered. This means the remaining columns are attached to this key at the leaf level of the index. In other words, once SQL Server has found the key, it has also found the row (to be clear, this is because of the clustered aspect).
An identity column is simply a method of generating a unique ID for a row.
These two are often used together, but this is not a requirement.