MY COMPANION HAS AN EXCLAMATION MARK INSTEAD OF A HEAD!
This is a rather common problem. When you unzipped the files, they ended up inside a folder in your data files section, right? And then you tried to drag each texture and mesh manually into their right folders? This doesn’t always work out as it should.
What you should do instead is this:

a) Create a subfolder on your desktop.

b) Unzip the mod to the sub-folder.

c) Doubleclick on the sub-folder.
Inside, you will find one meshes folder, one textures folder, one icons folder and an esp-file.
Take the meshes folder and DROP it into your Morrowind datafiles directory. It will automatically MERGE with your existing meshes folder (which will guarantee that all meshes ends up at the exactly correct places). You will get a message concerning this from the computer, say OK.
Now, take the textures folder and DROP it into you Morrowind datafiles directory.
Now, do the same with the Icons folder.
Finally, place the esp-file in your Morrowind datafiles directory.

That's all. You are ready to play your new mod.

I HAVE UPGRADED TO A NEW VERSION OF MY COMPANION, BUT IT HAS NO NOTICEABLE EFFECT.
This can happen if you upgrade your companion in your ongoing game. It’s caused by the fact that Morrowind sometimes stores an extra NPC-line in your saved game with data of your companion. What you should do is follow this step-by-step-instruction (by Fernelf) for cleaning your saved game:
a) MAKE A SAFETY COPY OF YOUR SAVED GAME!!!
b) Place the copy of your savedgame in the Morrowind data files folder. Rename your saved game from an ess-file to an esp-file (right-click on the save and hit rename.Exchange the letters ".ess" with ".esp", hit enter).
c) open up TES Construction set, highlight your renamed esp-file and mark it with "X". Hit the "details"-button (don't load the esp, just hit details).
d) In the details-list, look for and highlight a line called “NPC”+the ID-name of your npc, for instance "NPC-1AA_Morgana". DELETE this line. Use the "delete"-tab. You will get a question if you really want to delete the line, say OK to this. If you look in the details list, you will now see a "|" in front of the NPC-1AA_Morgana-line. This means that you have successfully deleted the line. Please note: there is also a line called "NPCC-1AA_Morgana". DON'T touch this line.
e) Close the TES Construction set (do not save or load the esp-file, just close TES construction set)
f) Go back to your data files folder and find the saved game esp. Right click and hit rename. Exchange the letters ".esp" with ".ess".
g) place your updated ess-file among your saved games. Now, you are ready to play.
Please note: this is rare, but some people may have to do one extra step, as they have problems to revert their new esp back to an ess. This is how to fix it:
go to “start”
hit my computer
hit tools
hit view and uncheck the last line "hide file extensions"
hit apply
now go back and now right click on the esp and hit rename and place the s where your p was.  

I DELETED MY COMPANION FROM MY ONGOING GAME. NOW I WANT TO ADD HER AGAIN, BUT SHE HAS NO HEAD!!
The reason for this is once again the “NPC”-line (see above). Clean your saved game – follow the instructions above – and your companion will get her head back.

I WANT TO GIVE MY COMPANION A NEW HEAD/A NEW RACE. HOW CAN I CHANGE THIS?
So, you have seen Elric Melnibone's beautiful Angelina Jolie face and want to change Laura Craft into Lara Croft? Or you have decided that Witchgirl Morgana should be a darkelf? Very well, that's easy to fix. Open up the mod in the constrution set. To the left, you will see a couple of tabs. One of them is called "npc". Click here. At the very top of the list, you should find my companions (I generally use the prefix 11A or 11AA). Let's say that you will alter Laura. Double-click on the line with her ID-name. That will bring up the NPC-infobox. Here, you can change her race, her looks and even her name (just DO NOT change her ID-name). To the very right, you will find a list of heads and hairs. Just pick your choice and hit the OK button. Her race is listed below her name, you can change her into anything you want. But, her voice will remain the same, as it is programmed in the dialog.
If you go to the Vivec-cell where she is (list of cells below the render window, there's a * at the Vivec cell where she is), you can see her change her appearance each time you click "OK" on her npc-info-box.
Once she looks as you want her to look, just save the mod. 
One thing, though: If you are already using Laura in your game, you will probably (but not always) have to clean your saved game in order to make the changes show up ingame (see instructions above). This should only take you a couple of minutes.

I WANT TO CREATE MY OWN COMPANION. HOW DO I ADD THE SHARING FUNCTION?
In the companion script, write:
Short companion
Set companion to 1
If you want to have your very own companion to tinker on, a good solution should be to download Grumpy’s excellent “Companion Project”, available here. This is a companion with the sharing function, a level-up system and the warping script, but with only a very basic dialog. You can change race, gender, head, hair to suit your own taste, and you can add all the dialog you want.  

I WANT TO DRESS UP MY COMPANION IN ARMOR AND WHIGS FROM METAL QUEEN BOUTIQUE, BUT SHE REFUSES TO WEAR WHAT I GIVE TO HER.
Hehe, you have probably choosen Morgana de Hex as your companion. Yes, she is incredibly stubborn, and she will only wear the very best armor. The reason is that her unarmored skill is much higher than her armor skills. Therefore, she is better protected without armor than with armor. Whigs are concerned as “light armor” with very little protection. Therefore, she won’t use whigs, either.
What you can do is this: Open up witchgirl mod in TESCS. Go to the npc-tab. Double-click on Morgana. To the left, you will find all her skills. Lower her unarmored skill and raise her armor skills so that they are higher than her unarmored skills.
If you are already using Morgana in your ongoing game, you might have to clean the saved game, following the instructions above, in order to update her skills. 

MY COMPANION HAS A TELEPATHY RING, BUT IT DOESN’T WORK ANYMORE.
Answer: You have left your companion behind for more than 72 game hours, which causes the infamous Bethesda original 72-hours-bug to execute. The game has now “forgotten” that your companion exists. As soon as you meet your companions again, the telepathy ring will “remember” them and function properly again. As for my newer mods (which includes the telepathy rings), there’s always a backup-solution to bring your companion back if you have lost track of him/her. Laura has given you an extra teleport ring, and in the Witchgirl Adventure, Morgana’s grandpa can help you. 

MY COMPANION HAS NO TELEPORT OPTIONS.
There exists a few mods that will still allow you to fast travel with your companion all over Morrowind, Solstheim and Mournhold. My VTA Travel Agency will make it possible for you to summon a travel agent who will take you and your companions to 32 various locations. CdCooley's Improved Teleportation will make all companions follow you when you use teleport spells. CdCooley's mod requires Fresh Fish's MWSE Script Extender, which is an external program that adds extra script functions to Morrowind.

I HAVE TELEPORTED MY COMPANION AND NOW HE REFUSES TO FOLLOW ME.
Sometimes, it happens that companions act odd after teleporting; they start shaking and won't follow. It's generally triggered by that you have either started to teleport yourself or started to teleport another companion before the companion has actually left the area.
There is a well working "cure" for this: 
Ingame, open the console and click on the companion so that his ID-name shows up at the top of the console. Type:
StartCombat player
This will most likely be enough - your companion won't attack you, just start to behave normal again. Should your companion actually attack you, just call up the console again and type:
StopCombat.

ARE THERE ANY OTHER GOOD COMPANION MODS AROUND?
Answer: Yes, of course!
In fact, there are nowadays so many that I cannot keep track of them all. 
My suggestion is to check out Telesphoros' List o' Mods.