If you don't have MS Word installed, then what App are you editing the document in? Remember, we can't see what you see, so we don't know what you are doing unless you tell us.
Usually you would want to have the Word app to edit Word documents. If you don't, it may be defaulting to a different app like Apple's own Pages app.
Sounds like it is opening the file in Pages which converts the file into a Pages document before opening it, and when you save it, it saves as a Pages document, not a Word Document. So it's understandable the recipient then can't open it on a Windows PC without the Pages App.
If this is true, then when saving the document in Pages, select the "Export To" option from the File menu instead of Save, and choose the "Word..." option to save it as an actual Word document.

Beyond that, you may want to get Microsoft Word if you are going to be sharing Word documents as Pages' translation is not always 100% accurate and there may be discrepancies specially in heavily formatted documents that use very specific Word formatting and set up.