I am a master student and I haven't published any paper yet. I have a manuscript of research but it recently got rejected. I wonder if I can put this in my CV (I might apply for a PhD) since it's empty in my publication section. Can I mark it as manuscript or something?
(Building on comments by @CameronWilliams and others under the question.) Many journals, including those in the Nature and Lancet groups, accept submission of a paper uploaded to a preprint service that they approve. So it's worth checking the journal websites. Also, if you submit to a journal you might not get the initial response, never mind reviews, for months. Uploading preprints to Arxiv, Medrxiv, Biorxiv, etc or to journal own preprint servers speeds up research dissemination and is pretty standard nowadays.
If you think your paper is worth showing to people, I'd put it to a good preprint service and link it up on your CV. Before that, it could make sense to accommodate reviewer's suggestions and others' feedback on your paper (eg, your instructor's, especially if you'd need a reference from them later).
If you are working to improve the paper or its underlying research with the hope of publishing it later, include a "Work In Progress" section on your CV and list the project there instead. Give the project a descriptive name and indicate that you have a paper in preparation. Don't say that it was already rejected, since it would be different in any future submission.
Having work in progress is an advantage in any academic application.