What is LocalDB?

It is a new version of SQL Server Express dedicated to developers to help them avoid a full installation of other editions of SQL Server.

Benefits.
Small installer. The 32-bit version has 28.2 MB and the 64-bit version has 33.7 .
Simplified. It does do not require configuration or administration.
Run as a low privileged user.
Simple installation.
Offers the same T-SQL language as SQL Server Express. It supports stored procedures,  geometry and geography data types, triggers, views.

Limitations.
It does not support Windows XP, Windows Server 2003, Window 2000.
It does not support WOW. LocalDB doesn't support installing 32-bit version on 64-bit Windows.
Allows only local connections. Only Named Pipes connections.
Only SQL Server 2012 Management Studio (on a computer updated with .NET Framework 4.0.2) can be used to manage LocalDB. SQL Server Management Studio of previous versions of SQL Server cannot be used.
Visual Studio 2010 RTM does not support LocalDB.

Read more at http://www.sqlcoffee.com/SQLServer2012_0004.htm