Simply replace the unfilled areas such as "" and "[date]" with your information.
Dear [name of supervisor],
I noticed on the listing of open jobs that there was a [name of position] open in the [name of department].
Please accept this [memo/email/letter] as notification of my intent to apply for that position.
Please know that I am very happy working for you in this department. However, I feel like the [name of position] will offer more opportunity and room for moving up. I have learned a great deal from you and my co-workers during my time in this department, but feel like it is time for me to move into other positions and build upon my knowledge and skill set.
Should I apply and am offered the position, please know that I will do anything I can to facilitate to smooth transition and assist in training any necessary replacement.