ArcGIS Online is a web-based product offered by ESRI, the company that also produces many other ArcGIS products, including Desktop, Server, and others.
Public accounts are available to any user, which have basic functionality and provide a chance to test some features.
The University of Michigan also has an ArcGIS Online for Organizations subscription through our site license. Accounts are created on an as-needed basis, and primarily for educational purposes at this time. The subscription is not limited in scope the way a public account is (see comparison), but the university has a limited number of credits for services. Access via http://umich.maps.arcgis.com/home/ if you already have an account.
ArcGIS Online Geocoding Services
ArcGIS Online Geocoding services are no longer free for use with ArcGIS Desktop or in apps. You may have used them previously, and many directions and tutorials assume they are available. You do not get a very clear error message when trying to geocode using these services.
To use the geocoding tools in ArcGIS Desktop, including ArcMap, you will have to use a file-based address locator or other geocoding service, if you have access to one. You can build your own address locator from reference data, or for US and Canadian addresses, the Streetmap North America data from ESRI includes street and zip/postcode level address locators. These can be accessed or copied from the ClarkData fileserver, at ESRI > Data_and_Maps_for_ArcGIS_10_1 > streetmap_na > data. This is only available to campus users.
Texas A&M Geoservices provides free geocoding services through a variety of tools, as well as other related services.
There are many other ways to batch geocode, but most are not free.