What type of yoghurt should I use for making curry?

I think for Indian recipes you should in general look for an unstrained, set yoghurt. There are other factors that determine the final taste and texture of the yoghurt (the bacteria, the type of milk, length of fermentation, …) but you may not have much choice w.r.t. other factors than these two:

Production process: Set yoghurt is yoghurt that's made the traditional way, it's fermented in the pot that it's sold in. Stirred yoghurt on the other hand is fermented before being packaged; it's a more industrialized production process which is cheaper, but results in a more sour, thinner yoghurt. Stirred yoghurt has a smooth, pourable consistency; whereas set yoghurt is firmer. I got a bit confused when I was using stirred yoghurt the first time I tried making raita and the recipe told me to whisk the yoghurt till it was smooth. The pretty sour taste of the yoghurt also didn't go well with the spices, and the juice from the cucumbers made the already-thin yoghurt too watery. You could try fixing both problems by straining the yoghurt a bit and adding sugar, but it's better and easier to just buy set yoghurt. The yoghurt's packaging might say explicitly what type of yoghurt it is, but if it doesn't or you don't understand the language: avoid yoghurt that comes in “milk carton”-like containers, that most definitely means it's stirred yoghurt. A plastic pot doesn't mean it's set yoghurt though, but it's a better bet.

Thickness: Strained yoghurt (like “Greek-style yoghurt” in Europe and the USA) is yoghurt from which the whey has been removed. It's pretty thick and sometimes also referred to as “yoghurt cheese.” In India it's used for example in the dessert Shrikhand, but if a recipe doesn't specify that you should strain the yoghurt, you should assume it's unstrained or briefly strained. Wikipedia says: “In south Asia, regular unstrained yoghurt (dahi), made from cow or water buffalo milk, is often sold in disposable clay pots. Kept for a couple of hours in its clay pot, some of the water evaporates through the clay's pores. But true strained yoghurt (chakka) is made by draining dahi in a cloth.” Should you really only find strained yoghurt, you can probably just thin it back with water or milk.

I should maybe add I don't have first-hand experience with what yoghurt from India tastes like, but I've observed that the ethnic shops in my area and one Indian take-away where I could peer into the kitchen all sell/use the exact same brand of yoghurt, which is an unstrained, set yoghurt with a mild (not too sour) taste.

Indian yogurt -- a brand called Desi Dahi (available in the US) is good -- will work well for what you are trying to do. Turkish or Greek yogurt may be ok as long as they are not non-fat or low fat versions. Whisk the yogurt well before adding. Also, be careful of overheating the dish after adding yogurt, because yogurt will curdle.

You can always make your own. I like the recipe from Show Me The Curry. The key here is to use Whole Milk as opposed to Skim or 2%.

The dahi/yogurt I use in my Delhi restaurant is made from full fat water buffalo milk.

It is probably more than 10% butterfat and has a consistency somewhere between heavy whipping cream and toothpaste (I'm not kidding, you have to squeeze this stuff out of the packet!)

If I have to use some crappy American yogurt I'd pick full fat Greek style. Although US yogurt is so full of other nonsense I hate using it.

But if you insist on using a 'low fat' yogurt in an Indian dish mixing a little starch in the low fat product will keep it from splitting & help it to 'thicken' like it is supposed to.

Some 'starches' I've used for example-

1 tsp gram flour (besan) per cup of yogurt- beat it in before adding, has a nutty flavor, 'expands' & thickens, gives a yellow hue

1 tsp corn starch per cup of yogurt- beat it in before adding, remove dish from heat before adding, stir well before returning to heat or may 'clot, does not interfere with flavor or color so it's great if you want a 'white sauce like for Safed Maaz

2 tsp white flour per cup of yogurt- beat it in before adding, can dull flavor a tiny bit or taste 'pasty if not well cooked, tends to 'lighten' the color of the dish

As far as Rogan Josh goes-

Rogan Josh is a Kashmiri dish, my husband is Kashmiri. I can tell you Kashmiri's would NEVER put tomatoes in Rogan Josh, In modern Indian 'Rogan Josh' just means any curry with mutton & a red sauce. The red hue of authentic Rogan Josh comes from Kashmiri mirch (Kashmiri red chills) and Ratan Jot (flowers of the cockscomb flower). But if you insist on making one of those less authentic 'Rogan Josh' recipes with tomatoes, add some starch to the yogurt before adding & it won't split.

The answer is in the fat content. Low fat will curdle when cooking.

I find Greek yogurt is just fine. Even low fat yougurt is ok, but only if you don't overheat it. When you add the yogurt, make sure the pan is off the heat and cooled down. Add the yogurt then heat gently and don't let it boil.

I personally think that you should aim for what you are comfortable with. The yoghurt from your region is fine either way. As long as you nail the curry seasoning you are incorporating the important part of the dish. There is a reason why fusion food is the most popular style. You cant find every ingredient every where. Capture the heart of the dish, add your own flair and try something new in every dish. After years of eating indian from true indian restaurants and at home with ingredients from the local indian grocer. Ive found many weird combinations that are in no way traditional but my family love them. My absolute favorite is my chicken tika masala, any hand fed starch and veggie with a strawberry greek yogurt red curry spice dipping sauce that sounds funny but is thr perfect blend of sweet savory and spicey. Tldr.... forget traditional experement have fun and make whatever you want, just call it a fusion and no one cares that its not tradional.

I love the turkish yogurts at our ethnic store. C-Layla brand for example. They even do an Iranian wild garlic yogurt which rewards the brave by being delicioous. Greek yog is also delicious. I suppose if you have any left you would want one that you could finish while cooking! My best advice though was taught me; add the yogurt (whatever brand) to the cooking pan by 1 tablespoon at a time, stirring it all the time clockwise until it disappears. Then add the next tablespoon and repeat etc. This stops any curdling or excessively milky meals.

Yogurt in marinades is great for tenderising meat. Be impressed by recipes that call for it. In my experience those recipes add a few spices too, possibly blended tomato and fresh coriander, and have excellent results