The Esri website has two flavors of user accounts: MyEsri and ArcGIS Online (AGOL) accounts. The main management site for these is accounts.esri.com.
This account is also known as your Esri account, ArcGIS Online public account, or Esri global login (the old name).
The MyEsri account is designed to be a single personal identity as an Esri user, not specifically identified with an organization, so it will travel with you through your life in GIS. This login is used to access your public (free) ArcGIS Online account, community.esri.com ("Geonet"). Esri site administrators from your organization log in to MyEsri and in turn add your MyEsri identity to the organization (yes, different from ArcGIS Online organizations!). The Esri site administrator than can grant you access to training, software downloads, and other items like User Conference passes. You can log in to MyEsri and then "connect" to any organization to which you have been added (if any) to access resources granted to you by your MyEsri organization's site administrators.
ArcGIS Online account
Also known as an AGOL organization account, AGOL organization login, or ArcGIS license portal.
Users normally have one login per each AGOL org they belong to. AGOL usernames usually have an organization suffix to help admins identify their users, and keep the usernames unique across ArcGIS.com. For example: jwpowell_USGS, N.Darton_SDMines. The AGOL login is used to access and use your organization's AGOL (subscription) site org.maps.arcgis.com, and also to authenticate named user licenses (currently, only ArcGIS Pro) from the arcgis.com or org.maps.arcgis.com license portals (either works, using your AGOL org username and password). Single Sign-On can be enabled by the AGOL administrator, if this is the case, AGOL users can login using their organization's credentials, and their username will look something like: firstname.lastname@example.org_orgmaps or user.name_orgmaps, where "orgmaps" is the AGOL organization name.
ArcGIS Online administrators have the ability to enable or disable Esri Access for each user. If Esri Access is enabled for that user their AGOL account can access MyEsri and the Training site with the same login credentials. Though this is convenient, if you ever are removed from your AGOL organization (new job, site re-configure etc) all your MyEsri information (contact info, training record, content) will not be available without asking Esri Customer Service to move it to a new (public) MyEsri account. Another advantage of keeping the accounts separate is you can participate in the GeoNet (community.esri.com) site without being associated with your organization, making it clear that you speak for yourself and not your organization! My opinion: it is best practice to disable Esri Access.
"User is not enabled for Esri Access"
If you get an error User is not enabled for Esri Access when trying to access your Esri login (for example to access training) follow the instructions given (these are much more useful than they were) to get logged in with your MyEsri account. If all else fails, shut down your web browser to log you out of everything, restart your web browser, visit my.esri.com and login with your MyEsri account. (If you don' t have a MyEsri account, create one.) New functionality has been added to the Esri signon system to "link" accounts which makes it pretty easy to switch among multiple MyEsri and AGOL accounts you may have.