I need help figuring out what is wrong with our Gaggia Espresso Maker. Here is a link to what is happening with it:

The pump seems to be working; it makes noise and the steam function works. I have tried priming it a few times to no avail. All the wires look like they are securely connected. It seems as though it will work but the water just drips out with no pressure. Has anyone experienced this? How do we fix it? Thanks for any suggestions you may have!

+1 on clogged grouphead. It looks quite dirty in the video.
Will it pump hot water through the wand? if yes, then +2 on the grouphead.

Probably also a clogged group valve too. While the dispersion screen is off, pull the group valve and clean it.
I noticed you still have an SPV. Pull the SPV return line out of the reservoir and put in a separate container and run the pump. See if you have a constant flow out of it. If you do, then your SPV has failed and needs to be plugged.
EDIT: SPV=Self Priming valve, located on the steam valve assembly, close to where it mounts on the top of the boiler. A silicone tube is attached, routed to the reservoir. This valve fails often, leading to brew water and steam flowing back into the reservoir, which shortens the pump life.

In the event you do not have them, here is a link to some manuals for your machine:

http://www.partsguru.com/GaggiaEspresso.html

Nice video.

Guys, Thank you so much for all of the help. The group head was completely clogged. I cleaned everything including the boiler and put it back together the water flow was WAY better. However, the pressure still didn't seem consistent. @ fwtechwiz: I think you were on to something about the SPV (self priming valve.) I didn't permanently plug it, but I took the hose off and plugged it with my finger with the power and espresso switch on. I replaced the hose and it's working great. Fwtechwiz, do you think it's important to plug it permanently?

Help! I have taken my Gaggia classic apart and can not unscrew the solenoid.
It stopped making coffee about 3 months ago and the problem was diagnosed as a dirty solenoid...I had descaled it a few times and that did not work...I finally got the nerve to open it up tonight and I have removed the solenoid but the "brass nut located near the bottom of the metal shaft" will not budge!
Any ideas? a few large, strong men have tried with a few different kinds of pliers...