The command you showed:

BACKUP DATABASE [DEVDB] TO DISK = N'C:\BACKUP\DEVDB_20121220_120038.bak' WITH NOFORMAT, NOINIT, NAME = N'UploadDownloadDev-Full Database Backup', SKIP, NOREWIND, NOUNLOAD, STATS = 10

GO

would put the backup on the SQL Server's 'C' drive. In order to back up to your local drive, the UNC name of the share on your local machine would have to be used, and you'd have to have rights to issue a backup command against that database, and I believe the account the SQL Server was running under would have to have access to that share.