EJB is a serverside component architecture that simplifies the process of building enterprise-class distributed componant application in
Java. By using EJB u can write scalable,reliable, and secure applications without writing ur own complex distributed component framework.
some examples where we can use EJBs are:
1. Stock trading system
2. Banking System
3. Customer call center
Where u can think of distributed working environment.
for the best start use following mastering ejb by ED Roman this one free PDF e-book. also this url contains some good articles related to EJB and its design
pattern http://www2.theserverside.com/books/masteringEJB/index.jsp