ArcGIS Online is a web-based product offered by ESRI, the company that also produces many other ArcGIS products, including Desktop, Server, and others.
Free Option with Limited Functionality: 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.
UM Option: The University of Michigan also has an ArcGIS Online for Organizations subscription through our site license. Accounts are created on-demand. First time users should go to https://umich.maps.arcgis.com and following the instructions on that page. 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. If you need ArcGIS for Desktop, you do not have to have an ArcGIS Online account, and you can find information about the campus license at http://www.itcs.umich.edu/sw-info/gis/arcgis.php .
If you already have a University of Michigan account, you can access it via http://umich.maps.arcgis.com/home/ .
To log into ArcGIS Online services via Desktop (e.g. ArcMap): Go to File --> Sign In. In the window that pops up, there is a link below the username and password boxes to "sign in with your enterprise login." Then on the next screen you will put "umich" into the box to complete the domain. The next window will ask you to choose between two options, and you should choose the first, "Using your University of Michigan account". This will then take you to the usual web-based login page for umich.edu and you can put in your uniqname and password. You will now have access to ArcGIS Online services that are available through the Desktop software.
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.
You can use ArcGIS Online Geocoding if you have a University of Michigan account (see above). In ArcGIS for Desktop products such as ArcMap, you will have to log in first in order for geocoding services to work. Geocoding in this way consumes credits, and may be slow and is inappropriate for large numbers of addresses (over 10,000).
Business Analyst for Desktop also includes geocoding services. Please contact email@example.com for more information about this option.
For large numbers of addresses, you might consider 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.