The .eu domain termination is a generic, hence it is not bound to geo-targeting on Google Webmaster Tool. In that sense, it is an alternative to the .com domain termination, if the .com is not available.
It was created by the European Union has a way to "communicate" that the business owning the domain has an European nature and that it is based on a nation of E(uropean) U(nion) and that is primary market is the EU.
From an SEO point of view, it doesn't offer any really advantage with respect any other generic domain name:
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you can't geo-target more countries with a single domain name
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you can't geo-target political regions (or continents).
Hence, it is good to have it for defending your brand, and to use it if .com (or .net) have been already taken. But if you have a .com, then it is better to redirect the .eu to it.