What is a Microsoft account?
A Microsoft account is a free account you use to access many Microsoft devices and services, such as the web-based email service Outlook.com, Office Online apps, Skype, OneDrive, Xbox Live, Bing, Windows, or the Microsoft Store. If you already use any of these services chances are you already have a Microsoft account. Microsoft account used to be called "Windows Live ID".
What is a Office 365 account?
If you have paid for a subscription to Office 365 either for personal use or as part of a school or work account, you will have an email and password to log into your Office 365 services. If the account is through school or work the email name will be set by an administrator. Read this guide from Microsoft for a more detailed explanation of their account types.