How do I make soft chapattis?

Yogurt and milk might work, but those ingredients are not in authentic chapati. When I hear the words "hard" and "rubbery" in reference to pan-fried bread, I'm almost certain that the problem is that they are cooking for too long on too low of heat. Increase the heat on your pan (it should be very hot) and cook the chapati very briefly on each side. This will still give them a nicely browned exterior, without turning them into sheets of rubber.

Oh well, I make chapatis everyday, even when I bake a Pizza or a cake.

The dough for making chapatis and stuffed and plain parantha and even poori is same. Until and unless you are trying some thing fancy like bhature you just need water and wheat flour.

As I am used to kneading dough w/out any measurement I cant suggest you measurements. I knead the dough till the time its firm enough to stand in a round shape. More over I take a ball of dough and throw it on granite counter with some force it doesn't loose its shape.

Now the temperature of griddle matters a lot. I keep it on a medium heat for chapatis and high for paranthas.
Roll out a chapati and place it on preheated griddle. You may need some flour while rolling out the chapati. Leave it on griddle for 10-15 seconds. Turn the side and leave for another 15-20 seconds. Turn side third time and you should see the chapati puffing up with steam. With this dough you can make chapatis, plain paranthas, stuffed paranthas and poori.

On a very special occasion my Mom used to knead the dough with milk and or ghee, a kind of clarified butter. Dough kneaded using milk and ghee is normally used for purees.
Apart from all the technicalities you may have practice a while. It took me a month to roll out a perfect round fluffy and soft chapati. Best of luck :)

Besides the tips above, you have to fold the chapatis a couple of times and flatten it. Like this :

This creates layers. Use a bit of ghee/pure ghee while folding. If you do this once or twice it and if the chapati is thin enough (2mm-3mm) it puffs up into a fluffy ball when you put it on the flame/pan.

One trick is to add a small amount of oil while kneading the dough. It makes the chapattis nice and soft.

Also, the more you knead the dough, the better the chapattis.

I don't know that unleavened bread is ever going to be super-soft, but are you simply cooking in a griddle as in the recipe Pointy linked? You can get them to really puff up if you expose them to direct heat. I'd try using two burners, one with a pan as before and one simply bare and over medium-high heat (experiment on this). After you brown both sides, use tongs to put the chapati directly on the burner -- it will inflate almost instantly, after which you can remove it from the fire. I've gotten quite good texture with this method.

Add good amount of milk or fresh yoghurt in place of water. It makes chapati soft

Please see About.com Indian Food Section @ Making Soft Chapatis (Indian flatbread)

I've been taught to roll them out three times. Roll out, spread with vegetable oil, fold into a bun, let rest while you do the same to the rest, then roll out again, spread with oil, fold back into bun, let rest. The last time you roll it out it is ready. Put in a medium-hot pan with oil and never let it sit. Keep spinning it with your fingers. Turn over and remove from the fire.

Adding some oil while preparing dough. And leave it for half an hour.

To get a softer and tastier chappathi.

Adding to the other answers, leave it for an hour or so, wrapped in a moist cloth. This would enhance the gluten formation.

Its all about how you add/maintain/remove moisture during the whole process of making a chapati.

I am not sure how anyone can measure the amount of water for the dough.

Use a little oil while kneading the dough, and add as much water as possible without losing the consistency you want.

Also, leave the freshly kneaded dough to settle for sometime (~15mins) by sprinkling some water all over the dough and cover it. The kneaded dough will absorb much more water during this time.

Once you have the rolled chapati off the tava, place it in a covered container/box.

I see there are several advices for preparing dough and rolling Roti posted on this site but I think an important trick while heating or cooking it is missing. So, I am adding an answer regarding heating/cooking it properly to avoid Roti becoming hard from my own experience for a week.

Note that this is not about making soft Bhakhri but it's for Roti.

Make sure that dough is soft. (Add little more water and after preparing, leave it for 3 to 5 minutes if dough is not soft enough). Also make sure that Rolled Roti is thin enough.

Heating or Cooking tip:

The key is: Roti becomes soft if it inflates completely.

• If you have just ignited gas stove, give some time to heatup tava. (Note that Metal Tava is suitable for Roti whereas Clay Tava is suitable for Bhakhari)
• Set the flow of gas fire to minimum before throwing rolled Roti on it.
• When rolled Roti is ready, drag it form rolling board/disk and drop on tava flipping it. (That means the face which is in front of you after rolling should be flipped and thrown on to the tava and now after throwing on Tava, opposite side should be in front of you)
• Wait only for 15 seconds and flip the side of Roti on Tava. (You'll very small inflated grains on cooked side)
• After flipping Roti, make the new side of Roti stayed and cooked in Tava for about 45 seconds. (Do not flip the side). Then increase the fire to maximum limit.
• Now, take Roti away from Tava and take Tava out of burner. Throw Roti directly on fire flipping it. That means you will flip the side before putting your Roti on fire.
• After throwing Roti directly on fire, you should see Roti starts inflating and it should inflated completely within 5 seconds. Now (even if Roti doesn't inflate in your case) immediately flip the side again to cook the opposite side just for another 2 to 3 seconds and take Roti out of fire. Done!
• Here the timing of 5 seconds when Roti flattens completely is very important since it makes Roti smooth. If you make delay then it will make roti hard.

I will try to capture video next time I prepare Roti and will attach pictures or gif.