I'm running Oblivion on high settings and I'm getting a good 40 - 50 fps. However when I play Morrowind on the standard settings, I am lucky if I get a solid 25.

My Laptop's specs are:

HP Dv6t Quad Edition
i7 Quad Core @ 2.0 ghz
6GB of RAM
ATI Radeon 6490M
and an Integrated Intel Graphics chip.

Also, the card isn't "switchable". There is software that you assign programs to run on which ever GPU you choose. I have Morrowind set for the ATI, but I can't seem to get it running smoothly. I can run Team Fortress 2 at 60fps on high settings and Fallout 3 on medium settings at a solid 60fps as well.

When I disable the integrated graphics I get a really awful resolution with many glitches. I'm new to PC gaming so I might be missing something pretty obvious. Is there a mod or something that utilizes Morrwind's engine better? Or is there something I can do to my PC to get it running better.

That's normal. Morrowind's engine is older and isn't as efficient.
I think there's something called FPS optimizer that can help you improve Morrowind's framerate.
Never used it myself, so I can only recommend you read all the instructions you can find on it.
Good luck!

The http://planetelderscrolls.gamespy.com/View.php?view=utilities.detail&id=22 is great for helping you stabilize FPS but it's not exactly going to boost your FPS.
It definitely worth trying though, it has great functions like
- Accelerate Mouse pointer in Menus: makes the mouse work better in menus
- You want to see farther: let's you control the Fog with your Mouse wheel, so you can adjust it depending on the situation.
- Hide objects that are currently small: Boosts performance by hiding small objects that are not near you. (and you can adjust the settings of how this is displayed)

Most people find better performance with the http://www.ntcore.com/4gb_patch.php on the exe. ( install after the Code Patch)

All that said, 25 FPS is totally fine. The game is playable at anything above 12 or 15.

edit:
A few other notes.
Run in Full Screen, not Windowed.
Close all other programs (like the internet.)
Disable shadows (in the in-game menu) They look ugly and hog FPS like crazy.

Open your Morrowind ini file and change this line: Max FPS=240 to Max FPS=60
This can slightly improve the performance, and definitively improves the functioning of some menu buttons.

Thanks for the help guys, but I'm having an issue when I try changing the FPS to 60, it say's "Access Denied" any advice on how to fix that? I tried turning off "read only" already.

Vista/7 have access restrictions on the Program Files directory. If Morrowind is installed there, is it strongly recommended to install it elsewhere.

All games should be installed elsewhere on either Operating System. Also Steam should be installed outside of the directory, if applicable.

So, anywhere that isn't Program Files?

Yes, although its generally recommended to put games under C:\Games, simply for easy access