In general, "alerts" refer to email notifications when something of interest to you has been published or added to a database. These can be on specific topics or from specific journals. Definitions, terminology and frequency of notification vary from publisher to publisher. These services are not available for all databases and journals, but are becoming popular features. Email notifications will usually require you to register your email address and preferences. Any email address will work, it does not need to be your official university email address.
Saved search features or subject alerts do not save the records you were looking at in your search; they save only the topic so it can be easily re-run to look for newer articles.
For more information about setting up alerts: http://www.lib.iastate.edu/services1/ref/tocserv.html
"My NCBI" is the feature within PubMed that allows you to save searches and set up e-mail alerts for new content. There are 3 short tutorials which explain how to register and how to save searches at: http://www.nlm.nih.gov/bsd/disted/myncbi.html