The procedure is a little different depending on whether you have Civ III 1.29f or if you are using Play the World.
Either way, you can hit the 'Add' button to add an entirely new unit, or you can hit the 'Rename' button to use a slot currently occupied by an existing unit. Assign the unit's values, costs, abilities, icon, etc. Also, designate it's Civilopedia entry (for example, PRTO_New_Unit, though I sometimes just copy the entry for a similar unit).
Next you'll need a unit folder that contains the unit animations (.flc files) and sounds (.wav files). It will also need an .ini file (most new units come with the .ini file, but if the unit you want to add doesn't just copy one from another unit, open it, change the appropriate .flc and .wav (or .amb) entries and save it. Be sure the .ini file has the same exact name as the unit's folder (I believe it is case sensitive too). For example, if your new unit folder is New Unit, then the .ini file should also be called New Unit.ini.
Another little trick is you can use sounds from other units by typing an entry in the unit.ini file. For example, say you wanted to use the ModernArmorFidget.wav sound for your unit, just type:
FIDGET=..\Modern Armor\ModernArmorFidget.wav
The unit folder should also have the same exact name that you gave the unit using the editor. However, this is where things differ somewhat. If you are using Play the World, you will need to add an entry to the PediaIcons.txt file that tells the scenario which unit folder to look for. If you scroll to the bottom, you should see something like:
#ANIMNAME_PRTO_Tank (this is the civilopedia entry)
Tank (this is the name of the folder the game will use for the unit)
So, you would need to add an entry for your new unit:
#ANIMNAME_PRTO_New_Unit
New Unit
The final step is to create an entry in PediaIcons telling the game which UnitsLarge and UnitsSmall.pcx files to use (I frequently re-use existing pictures for similar units), and you'll need to add an entry in the Civilopedia.txt file for the unit. You can just write a short description, it doesn't have to be anything fancy. Just copy and paste existing entries to get the format right and then change them where appropriate. These text files are located in the Text folder.
Say the new unit you wanted to add was a tank and you decided you just wanted to re-use the Large and Small pictures for the tank, your PediaIcons.txt entry would look like this:
#ICON_PRTO_New_Unit
art\civilopedia\icons\units\19shermantanklarge.pcx
art\civilopedia\icons\units\19shermantanksmall.pcx
Well, I hope this was easy enough for you to follow. There are also tutorials available on how to add units, so you can check out the tutorials forum for more info. Hope this helps.