scp works, but you can also use rsync:
rsync -e 'ssh -p <port>' <user>@<computer1address>:Desktop/test1.txt ~/Downloads/
Most of the time rsync is a drop-in replacement for scp, i.e. with most scp commands you can just replace scp with rsync and it will work the same way. The only reason that isn't true for your case is that rsync doesn't have a -p option to specify the port (well, it does, but it only applies when rsync connects through its own native protocol, not over ssh). So you need to tell it to use an alternate ssh command that includes the port.
If your ssh server uses the standard port, or if you have the port configured in ~/.ssh/config (in either case, this means you don't need to use the -p option when connecting with ssh), then you can just run
rsync <user>@<computer1address>:Desktop/test1.txt ~/Downloads/
In general, I prefer rsync to scp because it has many more options, and because if you're transferring a large number of files, rsync has the sense to skip ones that already exist on the destination system, cutting down on the amount of data that has to be transferred.
/s in your paths (as in "/home/remote_computer_user"). – grawity Jun 23 '11 at 19:25