The out-of-the-box functionality for uploading multiple files to a SharePoint document library relies on an ActiveX control that is only available on client computers that have the Microsoft Office system installed.
If the OOB approach causes problems for you, use my Multi Upload Feature for SharePoint instead. The feature adds an Upload Multiple Files command to the Actions menu of all document libraries. Once you invoke the command, a Click-Once Windows application is started right from the browser (no previous client-side installation required) that allows a user to select as many files as required from any drive connected to the client computer and upload those to the document library.