Like all industries, there are many words and definitions used in the training industry that people outside the industry may not understand. One of them is a Learning Management System (often called an LMS).
An LMS is simply a way of storing training records so that you know who has completed which training. A simple LMS is a spreadsheet with names down the page and courses and/or qualifications across the top with a tick or different colour when someone has completed the training requirements. More advanced LMSs will also tell you who requires the training.
Many LMSs now are online and have much more functionality. Some will allow you to deliver and record online training, webinars and other events and also allow the learners and their managers to enrol in training courses and events. These will send out emails, calendar invites and reminders automatically and make life much easier for the training co-ordinator.
But, the basic function of an LMS is always to store training records.