Hi,
dbo means you are logged in as the database owner. See: http://www.sqldba.org/articles/17-SQL-DBO-SQL-Server-Database-Object-Owner-dbo.aspx I assume that is why the server explorer is appending it to the database name.
A .mdf file stores data for a database. See: http://msdn.microsoft.com/en-us/library/ms189563.aspx
[Excerpt]
At a minimum, every SQL Server 2005 database has two operating system files: a data file and a log file. Data files contain data and objects such as tables, indexes, stored procedures, and views. Log files contain the information that is required to recover all transactions in the database. Data files can be grouped together in filegroups for allocation and administration purposes.

Database Files
SQL Server 2005 databases have three types of files, as shown in the following table.
| File |
Description |
|
Primary |
The primary data file contains the startup information for the database and points to the other files in the database. User data and objects can be stored in this file or in secondary data files. Every database has one primary data file. The recommended file name extension for primary data files is .mdf. |
|
Secondary |
Secondary data files are optional, are user-defined, and store user data. Secondary files can be used to spread data across multiple disks by putting each file on a different disk drive. Additionally, if a database exceeds the maximum size for a single Windows file, you can use secondary data files so the database can continue to grow.
The recommended file name extension for secondary data files is .ndf. |
|
Transaction Log |
The transaction log files hold the log information that is used to recover the database. There must be at least one log file for each database. The recommended file name extension for transaction logs is .ldf. |
[/Excerpt]
In your scenario, this is my understanding of the differences:
1) MyLocalHost\sqlexpress.Testing.dbo -- here the connection string looks like:
Data Source=MyLocalHost\SQLEXPRESS;Initial Catalog=Testing;Integrated Security=True;Pooling=False
2) Database.mdf -- here the connection string looks like:
Data Source=.\SQLEXPRESS;AttachDbFilename="C:\...\My Documents\Visual Studio 2008\WebSites\Testing\App_Data\Database.mdf";Integrated Security=True;User Instance=True
In instance 1, you are connecting just as you would over the network to another sql server. In instance 2, you are connecting directly to a database file locally, isolated from any "real" databases.
You can copy mdf files from machine to machine and later import them into a Sql Server installation using Management Studio.
In summary, the difference: in scenario 1, you are connecting as you would over the network, in this case, to localhost. in scenario 2, you are connecting directly to an isolated database file, an .mdf file.
I hope the explanation was clear. Maybe someone else will correct me where my understanding is wrong. But I thought you might at least appreciate the answer above since no one else seems to be replying.
Pete