Yes and no.
There are two primary ways to do this.
The first is known as "cross listing". Basically, this means that shares are listed in the home country are the primary shares, but are also traded on secondary markets using mechanisms like ADRs or Globally Registered Shares. Examples of this method include Vodafone and Research in Motion.
The second is "dual listing". This is when two corporations that function as a single business are listed in multiple places. Examples of this include Royal Dutch Shell and Unilever. Usually companies choose this method for tax purposes when they merge or acquire an international company.
Generally speaking, you can safely buy shares in whichever market makes sense to you.