How to execute T-SQL for several databases whose names are stored in a table?

The simplest way is this:

DECLARE @stmt nvarchar(200)
DECLARE c CURSOR LOCAL FORWARD_ONLY FOR SELECT 'USE [' + Name + ']' FROM DatabaseNames
OPEN c
WHILE 1 <> 0 BEGIN
    FETCH c INTO @stmt
    IF @@fetch_status <> 0 BREAK
    SET @stmt = @stmt + ' ' + @what_you_want_to_do
    EXEC(@stmt)
END
CLOSE c
DEALLOCATE c

However, obviously it will not work for statements that need to be the first statement in a batch, like CREATE PROCEDURE. For that you can use SQLCLR. Create and deploy a class like this:

public class StoredProcedures {
    [SqlProcedure(Name="exec_in_db")]
    public static void ExecInDb(string dbname, string sql) {
        using (SqlConnection conn = new SqlConnection("context connection=true")) {
            conn.Open();
            using (SqlCommand cmd = conn.CreateCommand()) {
                cmd.CommandText = "USE [" + dbname + "]";
                cmd.ExecuteNonQuery();
                cmd.CommandText = sql;
                cmd.ExecuteNonQuery();
            }
        }
    }
}

Then you can do

DECLARE @db_name nvarchar(200)
DECLARE c CURSOR LOCAL FORWARD_ONLY FOR SELECT Name FROM DatabaseNames
OPEN c
WHILE 1 <> 0 BEGIN
    FETCH c INTO @@db_name
    IF @@fetch_status <> 0 BREAK
    EXEC exec_in_db @db_name, @what_you_want_to_do
END
CLOSE c
DEALLOCATE c

You should be able to do this with the sp_MSforeachdb undocumented stored procedure.

This method requires you to put your SQL script to be executed on each DB in a variable, but should work.

DECLARE @SQLcmd varchar(MAX)
SET @SQLcmd ='Your SQL Commands here'

DECLARE @dbName nvarchar(200)
DECLARE c CURSOR LOCAL FORWARD_ONLY FOR SELECT dbName FROM DatabaseNames
OPEN c
WHILE 1 <> 0 BEGIN
    FETCH c INTO @dbName 
    IF @@fetch_status <> 0 BREAK
    EXEC('USE [' + @dbName + '] ' + @SQLcmd )
END
CLOSE c

Also, as some have pointed out. This approach is problematic if you want to run a command that needs to be the only thing in a batch.

Here is an alternative for that situation, but it requires more permissions than many DBA's might want you to have and requires you to put your SQL into a separate text file.

DECLARE c CURSOR LOCAL FORWARD_ONLY FOR SELECT dbName FROM DatabaseNames
OPEN c
WHILE 1 <> 0 BEGIN
    FETCH c INTO @dbName 
    IF @@fetch_status <> 0 BREAK
     exec master.dbo.xp_cmdshell 'osql -E -S '+ @@SERVERNAME + ' -d ' + @dbName + '  -i c:\test.sql'
END
CLOSE c
DEALLOCATE c