There are two ways that I apply textures to my models: The Texture Coordinate Editor(TCE) or creating a UV map with LithUnWrap. Using the TCE is easier, but doesn't always give the best result.

Assigning Textures

Using the Texture Coordinate Editor

Deselect all, and under the "Groups" tab, select a group and then look under the "Materials" tab. Click "new" and then click one of the buttons labeled "none". Select a texture; I'm using one from the Construction Set CD. Then click "Assign". If you look at the model in textured mode (right click in the window) you should see the texture now.

TIP: You can also use other textures if you have them in your Morrowind\Data Files\Textures directory. It's best if they are sized in powers of 2---128, 256 and 512 pixels in width and height. You could try 1024 or larger but it may not work. Save them in TGA or DDS format. BMP may work and it may not. Other formats such as JPEG won't work at all.

Using the Texture Coordinate Editor

When you have assigned textures to the groups, select all and press ctrl+T. This will open the TCE. In the second roll-out, pick "Top" and then click "Remap". Then set the first roll-out to the other group and click "Remap" again. This will line the textures up with the tops of the groups. When you are done with the TCE click the window close. You can make the texture repeat with the "Tile Texture Mapper" under Tools.

Making a UV map with LithUnWrap

An alternative way to texture the model os to make a uv map using LithUnWrap. I save my model and open LithUnWrap. Under File>Model>Open, I pick my file. Then I go to Select and pick "All". Then I go to Tools>UV Mapping>Decal and pick the y axis alignment.

This is the result:

There's a lot of wasted space, so next I set the selection to groups (Select>group) and move and scale the parts so that more space is covered. Right clicking opens up a menu of options like scale, rotate ect.

You can select individual triangles and move them, to spread the shapes over a wider area, but I generally don't because that makes it confusing to edit later.
When I'm finished arranging the groups, I go to "File>Export>UV..." and export the model in .luv format. And I also go to UV Map>Save and save the map as either a .bmp or .tga.
Back in MilkShape, with the axe model loaded, I import (File>Import>Lithium UnWrapper LUV...) and pick the .luv file I made in LithUnWrap. There's always a message about welding the groups if needed. Most of the time it's not necessary so just click "ok".

Then under the "Materials" tab I make a new material, select the UV map I made in LithUnWrap and click "Assign". In Textured mode the model should have the map applied to it.

Because this is just a tutorial, I'm not going to do anything with the texture, but usually I open it up in a graphics program and color it in. Basically I select areas and fill them with a gradient blend or a pattern. The Gimp has some nice metalic blends and patterns. I also "dirty" it up with filters like "erode" or "Van Gogh (LIC)".
There's a useful plugin for MilkShape called "Refresh Textures" that comes in handy at this point. While I'm working, I can view my progress in MilkShape with this tool. It adds the command "Refresh Textures" under the Edit menu.