Family Tree DNA is based in the U.S., but has customers worldwide -- they can and do ship DNA kits anywhere. They also have resellers in Europe and the Middle East, which helps bring in the international samples.
Other U.S.-based companies only ship within the U.S. or have limited international reach. For example, Ancestry.com's new AncestryDNA test is U.S.-only at the moment, and 23andMe ships to a limited list of countries.
There are genetic genealogy companies outside the U.S., but the biggest ones are U.S.-based.
The International Society of Genetic Genealogy (ISOGG) has a list of genetic genealogy companies at their website.
All companies have their own databases, where you are compared to other customers of the company. However, depending on what type of test you are doing, there are also external databases where you can upload your results for further comparison. Some companies also allow uploading results from another company.
Disclaimer: I'm currently a consultant for FamilyTreeDNA, though I started as a customer first in 2005.