The major strength of the European Union is in the fact that it creates a very large market for all of its members. The EU countries can trade with one another without limitations, just as if they were one big country. This creates a larger market which can work more efficiently than many small national markets could.
The major weakness of the EU is mainly political. The EU is made up of many different countries with different political cultures and different levels of prosperity. All have to agree on various policies. Today, this weakness is being seen in the response to the Greek crisis. Germany in particular wants other countries like Greece to be more fiscally responsible in return for help from the EU as a whole. Greece and other countries feel that Germany is being too demanding. These political differences make it hard for the EU to stay together as a single entity.