All three can be used on your desktop (without an Internet connection, if necessary) and they all have an online version that can be synced with the desktop version.
They function similarly, so you can always transfer citations between EndNote, Mendeley, and Zotero. That said,
EndNote Desktop has been around the longest and is generally more robust in terms of customization, i.e., Endnote citation styles are more easily customizable than Zotero's or Mendeley's. A drawback to Endnote Desktop is that it won't be available after graduation and constantly requires updated purchases. (Endnote Web doesn't have optimal functionality on its own without the companion desktop version.
Mendeley's strength is its ability to import and grab metadata from pdfs (including pdfs already on your computer). It was also created as a collaborative tool and has more social media features than the other two tools. However, only the basic version (with much less functionality) is free.
Zotero is by far the best at "grabbing" citations and full text from catalogs, databases, web pages, and sites like Amazon and Flickr. It’s free and open-source with a large user group, user forums, and a developer community, and will be available to you after graduation. Syncing your citations to Zotero's cloud is seamless.
The EndNote software can only be used by current KAUST University staff or students.
The software may be installed on University-owned computers and personally-owned computers, including home computers and laptops. The computer owner must be a current KAUST University staff member or student. The use of the program must be for coursework, research, or other work directly related to KAUST University.
The software may only be used during the term of KAUST University’s license agreement.
If the University ceases the site license or if you leave KAUST University, you must discontinue use of and uninstall all copies of the Software. No further product upgrades or technical support will be provided once the license has ceased.
The software cannot be sold, lent, or transferred to anyone or electronic forum except eligible members of KAUST University.
The software’s component parts may not be separated for use on more than one computer. The following is also prohibited: reverse engineering, decompiling, translation, modification, distribution, broadcasting, dissemination, or creation of derivative works from the Software.