Mother Church Mouse is, well, a mouse.  She is the wife of Father Sexton.  They both live in the wall of the church of Nottingham, and assist Friar Tuck with its administration and upkeep, and ministering to the common folk of the village.  She has a general American accent.

We never specifically learn her name, but we do get to see something of her personality.  While she is a sweet, normally gentle and kindly mouse, who is even willing to give her very last farthing* for the church poor box (which the Sheriff of Nottingham regularly raids), she is also quite capable of being blunt and outspoken, and is quite brave enough to speak her mind when she feels it is necessary.

Normally, however, she contents herself with keeping the church in as good a shape as she can manage, sweeping, dusting and generally picking up.

Origins of the character: While there are no specific foundations for this character in the Robin Hood legends, Mother Church Mouse does represent a sample of the average citizens of the village of Nottingham.
 

*It's worth noting that, in historical terms, the farthing coin was not first minted until after the deaths of King Richard and Prince (later King) John.  At least in historical terms.  Not terribly important from a movie perspective.