Rotation is just shifting with the bit exiting from one end fed back into the input at the other end, possibly by way of the carry flag bit. At the level of a simple implementation, it would make sense to have one circuit for both operations, with some additional control lines to select the source at the input side between the output of the other side, 0, or 1. Sign extension during right shift of 2's complement numbers would be another selectable option often built in.
The stackexchange sites aren't really suited to "list" questions including about applications, but a couple come to mind:
If you want a vector to test every bit of another value in turn, and to do so repeatedly, you could just keep rotating an initial one-bit-active value over an over, never having to re-initialize it.
You could swap a two-part (typically double-byte) value to imitate the opposite endieness of encoding by rotating it half way. Or putting it another way, it can be a single-operation swap of the values of two pairable but also independently accessible registers (think AL and AH together making up AX in real mode x86). But this will not work two endian swap a four-part value, such as a 32-bit value on a byte-addressable machine.
Various coding, checksum, and hashing schemes may wish to transform a value
