When at that prompt type in
sudo apt-get install apt

Enter your password that your logg into your splash screen and it should fix it.
You may have removed some thing that you should not of. The system is telling
you what to do, but it happens when it can be very vague.

He brought that up.

You're using apt to install apt, which doesn't really work.

Have you tried installing it with synaptic?

Synaptic is just a graphical front-end for apt.

Not sure if aptitude uses apt or if it's a separate package
installer... it seems to be on my system, so I guess it's on there by
default. Should be worth a shot.

If not, you may want to look at finding source code for it and
installing from that - I don't think you need apt to compile. However,
I did not dig up any source code for it after a bit of a Google search
and you may also run into the problem that you don't have the required
files to compile it. In that case, the best I can tell you is make sure
your data is backed up, and reinstall the system.

https://wiki.ubuntu.com/Home?action=fullsearch&context=180&value=APT&titlesearch\
=Titles

which apt is now the question....

I would recommend you start troubleshooting by:
grep "apt-get" /var/log/*
and see which messages appear.

You might try starting synaptic and look at the status bar at the bottom to
see if there are any "broken" packages.