Copy files from a hard drive onto an iMac? I had an old Mac 5.1 that now, sadly, broke down. Now I still have the SSD, can I completely copy everything (including OS etc.) onto the new iMac I bought? I have an USB/Hard Drive connector.
I also don't want to completely replace the new SSD in my new iMac, because it has more capacity.
You have different options:


Use Migration Assistant and select the option to transfer from a Mac, Time Machine backup, or startup disk. This method is probably most convenient, however, as noted in a comment by user Tetsujin: "[...] unless you migrate as the very first step in setting up a new machine, you will end up with two accounts - the one you just set up &amp; the migrated one. It is possible (if irritating) to merge them, but if the new machine is still pretty much a blank canvas, it could be simpler overall to nuke it &amp; start over."
Boot the old SSD from the USB enclosure by pressing alt during boot and selecting it. Then use a software such as Carbon Copy Cloner or SuperDuper! to copy the entire partition.
Alternatively, you could also use Disk Utility instead of Carbon Copy Cloner or SuperDuper! as described e.g. here (edited from the source):



  
  Launch Disk Utility
  In the sidebar, select the volume to which you want to copy/clone data. The volume you select will be the destination drive for the Restore operation.
  Select Restore from Disk Utility’s Edit menu.
  A sheet will drop down, asking you to select from a drop-down menu the source device to use for the Restore process. The sheet will also warn you that the volume you selected as the destination will be erased, and its data will be replaced with data from the source volume.
  Use the drop-down menu next to the "Restore from" text to select a source volume, and then click the Restore button.
  The Restore process will begin. A new drop-down sheet will display a status bar indicating how far along in the Restore process you are.