Robert Chambers wrote about this verse in his book called Popular Rhymes, Fireside Stories and Amusements of Scotland: "One of the first whimsicalities practiced was to take the two feet of the infant, and make them go quickly up and down and over each other, saying the following appropriate verses…"

Feetikin is the diminutive of feet... so here it means "little feet".

Feetikin, Feetikin - Scottish Children's Songs - Scotland - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Thanks and Acknowledgements

This nursery rhyme can be found in The Big Book of Nursery Rhymes (circa 1920) edited by Walter Jerrold (1865 - 1929) and illustrated by Charles Robinson (with some graphical editing of the above image by Lisa Yannucci).