iMac 27" late 2009 running OSX 10.10.4 - please explain time machine to me?
Two TB should be more than enough. It's a good idea to start with an empty HDD, so if you had any data on it; that could be the problem is. It could also be because the drive was configured for Windows OS. 

If you want to use the HDD as time machine and to store other stuff you should partition it. It's simple just open Disk Utility, select the disk, go on partition tap, and the plus button. Make sure the format is Mac OS Extended, then make the new partition of about 1.5 TB. After it's done, you should see two drives connected. The old one (0.5 TB) should still contain all your data from before if it was less than 0.5 TB. You can just format the new one if you like. Then just restart the Time Machine process; from Time Machine Preferences Pane > Select Drive. 

You also configure which file you don't want to backup, things which you can just download again, from the options button in Pref Pane.