Enlargement is the process whereby countries join the EU. Since it was founded in 1957, the EU has grown from 6 member countries to 27 .
Welcoming new members was part of the plan from the beginning. The founding fathers were confident enough of their idea to leave the door open for other European countries to join.
Helping countries that have the potential to become members has been the EU’s response to changes in the European political landscape over the past 50 years, promoting economic growth and strengthening democratic forces in countries emerging from dictatorship.

Romanians celebrate joining the EU on 1 January 2007.
The 6 founding members of the EU in 1957 were Belgium, France, Germany, Italy, Luxembourg and the Netherlands.
From 1973 on, most of the other Western European countries joined.
Then, following the collapse of their regimes in 1989, many former communist countries from central and eastern Europe became EU members in 2 waves, between 2004 and 2007.
The Treaty on European Union states that any European country may apply for membership if it respects the EU's democratic values and is committed to promoting them.
But specifically, a country can only join if it meets all the membership criteria:

Turkey is a candidate to join the European Union.
The process has 3 stages (all subject to approval by all existing EU countries):
When the negotiations and accompanying reforms have been completed to the satisfaction of both sides, the country can join the EU – again, if all existing EU countries agree.
Currently the EU has offered the prospect of membership to 9 countries: Albania, Turkey, Iceland and all the countries of the former Yugoslavia (except Slovenia, already an EU member).
5 of these have been granted official candidate status:
Croatia
is an acceding country, expected to join the EU in mid 2013.