Problems Connecting to Database With ODBC

I am a total newbie to SQL Server, so I am a little lost.  I have a SQL Server 2005 database on server A.  Because Server A is being decommissioned, I had to copy the database to server B, which I have done (both have SQL Server 2005 installed).  I am able to log into the new database with no problems etc.

I have a separate client server which has to connect to the new database using ODBC.  When I try and connect to the database using ODBC, I get an error stating that the database cannot be found.  The database is named exactly the same as on the old server.  I am able to connect to the old database using the same client machine, using the same database name with no issues.

Am I missing something?  Like I said, I am brand new to SQL Server.  This is the first time I have tried copying a database.  Thanks everyone.