In a very basic way, these terms are interchangeable. But, in modern society, they tend to have slightly more distinct meanings. In general...
A choir is a group of singers who primarily sing religious music. The word choir is also used to distinguish a subset of a larger ensemble (the 'woodwind choir' of an orchestra, for example).
A chorus is a group of singers who primarily sing secular music. This term is often used to describe the "backup" singers in plays and musicals.
The term chorale is slightly outdated, but is more synonymous with choir, in that a chorale typically sings religious music. (Incidentally, 'choral' is the adjective used to describe music written for a vocal ensemble to perform.)
It doesn't matter if the singers are accompanied by musicians or not - the meanings of these words doesn't change.