What is the difference between a lager and an IPA in the brewing process?
The main difference is the yeast, Ale is brewed with a top fermenting yeast s.cerevisiae whereas Lager is brewed with a bottom fermenting yeast s.pastorianus. From this comes the fermentation temperature, ales are fermented at higher temperatures(14-20 C) than lager(10-12 C).
A lager you would also allow to warm towards the end of the primary fermentation for a couple of days for a Diacetyl Rest, this is not required with an Ale.
Fermentation of a lager takes longer than that of an ale due to the lower temperatures.
Lagers should according to their name then be stored for a period of time in secondary/lagering vessels. This is not always required and a drinkable lager can be produced in under 2 weeks, but I, personally, would question if it should be called a lager if it has not been stored at 0-4 deg C for a week or more.
Now to be more specific and answer your question regarding IPA vs Lager:
Lagers evolved initially in Bavaria and later in Bohemia with Pilsner, in the early 1800s. IPA evolved in London then later Burton. Both these Paler varieties of what had gone before where made possible by new indirect kilning methods allowing the production of lighter malts.
This little bit of history is important as lager is usually made with a double or triple decoction mash, where as IPA are generally made with the more British single temperature infusion mash.
There are no hard and fast rules though. You could technically make either with either method, and these days many lagers are made with single infusion and I am not aware of any IPAs made with decoction.
The amount of hop bitterness is another big differentiator IPA are often highly hopped >40 IBU, commonly >60; where as lagers are in general far more subtly hopped. (20-40 IBU)
Simply put, there are two overarching umbrellas in beer... ales and lagers. An IPA is an ale. GENERALLY speaking all beers fit under one of these two umbrellas. Once under one of these umbrellas the main difference is brewing technique. Even if you use the exact same ingredients and technique between a beer with ale yeast and the other with lager yeast, one is still an ale and the other is still a lager. You would rarely use the same technique between the two but that's not the point here.
What can make this a confusing concept to learn is that lager is not only a description but can also be a verb whereas ale doesn't have this usage. An ale is just a description. To "lager" (verb) is a process of conditioning beer (AKA letting it sit) at low temps for an extended amount of time. You can lager any beer, even an ale. Furthermore, you don't necessarily have to lager (verb) a lager (descriptor). The classic example is Anchor Steam which originally was a lager (because it uses a lager yeast) brewed at higher than normal temps, though many people nowadays still apply some level of lagering.
In short, an IPA is an ale and can never be a lager. If you use the same ingredients for an IPA but use lager yeast and then lager it, it becomes an IPL.
There have been some good answers about lagers vs ales. I’m going to expand on the IPA (Indian Pale Ale) side of things. Obviously by the name it’s an Ale. What makes this Ale ‘different’ is the way it’s hopped and essentially the story of its origin. In the 1800s there was export of beer from England to The Indian colonies. This beer was all Pale Ale. The trip to India took a while and it was hot in India so beer didn’t have a very long shelf life at all. However over time one style of beer seemed to last longer than other styles - it was found that in the making of this style a large amount of bittering hops was used. Without realizing it the increase in Hopps or alpha acids had a preservative effect. It eventually became a popular beer in the Indian colonies and thus got the name Indian Pale Ale.