Unable to lock the administration directory

Receiving the following error when trying to install gimp:

E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied)
E: Unable to lock the administration directory (/var/lib/dpkg/), are you root?

I am fairly new to Ubuntu and understand a Synaptic process is running. How can I tell in the System Monitor which is a Synaptic process.

This is on 10.10.

You have another problem

To see the synaptic application running in System Monitor; just go to the tab Processes and (preferably) sort by Process Name. Synaptic might be in that list. You can also click on one of the entries and start typing Synaptic to automatically focus on any Synaptic Process.

Please note: it is also possible that another user or the root user is running Synaptic, make sure to set View All Processes in the View menu to be able to view everything.

But honestly: this error message is not caused by another process running, but by running synaptic as an unprivileged (regular) user. You need to use the root user account to run synaptic or apt-get or aptitude to install software. That gives the error message you get (and it also hints towards being root). So in the graphical interface:
gksudo synaptic
This is (approximately) what is executed when you use the menus: System > Administration > Synaptic Package Manager

in the terminal you better use one of the following:
sudo apt-get
sudo aptitude

In case another apt-client (synaptic, apt-get aptitude, ...) is running, you will get the following message:
E: Could not get lock /var/lib/dpkg/lock - open (11: Resource temporarily unavailable)
E: Unable to lock the administration directory (/var/lib/dpkg/), is another process using it?

Another possible explanation

Another possible explanation is that all forms of package management - dpkg, apt, aptitude, gdebi, synaptic etc - use just one database.

So you can only use one package manager at a time. Close all open apps. Use Synaptic just by itself.

You can install Gimp from the Terminal. (Be sure that Synaptic is not open!). Just use the command: sudo apt-get install gimp