The purpose of mortar is to discourage the stones from moving.Options
Many of the ancient Anasazi buildings (in the Southwestern United States) were built using cob or earthen mortar, along with a lot of attention and skill in placing the stones. Some of these earthen mortars have a little lime added to them.
Some ancient sites and many European walls were built using lime mortar between the stones. This was basically lime mixed with sand. You'll have to do some research to find out more about lime mortars. I have never used them so I can't give you much help.
This is the modern version of earth or lime based mortars. It consists of sand, lime, cement and water.
Concrete and rebar wall with a stone facade
This is what stone walls have devolved to. The library will have books on how to make these modern fake-stone walls.
The recipe is as follows: one part premixed mortar cement to three or four parts sand of various sized particles, and water. (You can make your own mortar cement by using 2/3 portland cement with 1/3 dry builder's lime instead of premix.) Use three parts sand if you follow instructions well, four if you are conservative and want to stretch the cement a little farther. The production of cement is very bad for the environment so the less you use, the better. Caution: cement burns skin and eyes! Wear gloves and eye protection.
When you make up the mortar, mix small batches and use immediately. First mix the dry ingredients well, then add the water. It will start to cure as soon as the cement gets wet. Mix it in a wheelbarrow using a hoe or trowel, or on a piece of plastic or tarp by shifting the stuff back and forth.
You can use a wet mortar, or a dry one. When making a wet mortar, watch out
because it will go from too dry to too wet very quickly. Add the water carefully
until the mix makes stiff peaks. If it does get too wet, it's OK to add a little
sand to stiffen it up. When using a dry mortar, barely moisten your ingredients.
Mix well. Then it's ready to use!
tools you'll need for mortar
Squish mortar into the cracks between the stones you've already set, wiggling it in with a trowel or stick until the holes are full. You can add small stones too. Do this after each course of stone is laid. Clean your tools, gloves, and wheelbarrow before the mortar hardens on them.
For a dry mix, barely moisten the ingredients and wet the surfaces of the stones generously before troweling the mortar into place. If you are using a dry mix for your foundation, you may decide to make up a wet mortar for the places between the stones on the very outside surface of the foundation - the dry mix tends to fall off the sides of the foundation before it cures.
Cover the fresh mortar with a tarp or with damp burlap bags.
If you've laid the mortar in the morning, you can come back in the evening or the next day and tidy up any joints and stone faces that will be visible. A wet whisk broom will do the job unless the mortar has hardened too much, in which case use a wire brush.
Keep your mortar work covered for a week or so. Spray water on it every day so it doesn't dry out too fast. This is very important to the strength of the mortar.
When you add more stone to a partially cured section of the foundation, be gentle so you won't dislodge yesterday's work.