ArcGIS Online is a web-based product offered by ESRI, the company that also produces many other ArcGIS products, including Desktop, Server, and others.
The Get Started with ArcGIS Online is a great place to start, and will give you an account for learning purposes. Public accounts are also 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. The subscription is not limited in scope the way a Learn GIS or public account is (see comparison), but the university has a limited number of credits for services. Send an email to firstname.lastname@example.org with a description of how you plan to use your ArcGIS Online account, and Peter (LSA) or Nicole or Mara (UM other than LSA) will get back to you with more information.
If you already have a University of Michigan account, you can access it via http://umich.maps.arcgis.com/home/ .
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.