Use "kipkipin" when you want to tell someone to place something pressed between his side and his arm.
"Kipkip" is the root word, but is often used also as its simple present and present progressive tenses. "Kipkipin" is an object-focused verb.
Kipkipin mo na lang itong libro para libre pa rin ang parehong kamay mo. = Just carry/hold this book under your arm so you can still have both your hands free.
Ano 'yung kipkip ni John? = What is John carrying under his arm?
Ano 'yung kinikipkip ni John? = What is John trying to carry under his arm? ("kinikipkip" would be its present tense conjugation. It may be used in place of "kipkip" in the preceding example, but used more often to mean the act itself of trying to keep an object pressed between one's side and his arm.)
Huwag kang magkipkip ng diyaryo para hindi mamantsahan ang puting kamiseta mo = Don't carry a newspaper under your arm so as not to stain your white shirt. ("magkipkip" is a subject-focused verb).
There's another word that uses the root "kipkip". That word is "pakipkip", a noun. If you get invited to baptisms, you might just hear the word. It refers to the monetary gift, usually placed in an envelope, given to the infant. Of course, we all know that the child is just a few weeks old and therefore has no idea as to the wealth his/her parents are amassing through him/her. :) Well, that's how that custom works. Customarily also, the "pakipkip" is handed over to the parents covertly, if possible.