What is the difference between these sentences?
1 In the past I would eat a lot of appples
2 In the past I I might eat a lot of apples
1 is a repeated action I ate many apples often in the past. How would you define 2?
"Would" has multiple functions, so sentences with "would" can often be formally ambiguous. Here is seems to be the "past habit" sense of would, and means "I used to eat a lot of apples".
The second sentence is odd. A more realistic example could be "In the past, teachers might beat students with a cane". Some teachers did. It is possible that a particular teacher used to do this, but not all teachers did. It's not the best sentence, and probably could be rephrased to express the idea more clearly. Your sentence with apples doesn't really work. But context can help, as in the comments "In the past, I might eat a lot of apples when they were in season, then not touch one for months." (Sometimes I did, but not every season)
You could say "I might have eaten a lot of apples". This expresses something in the past that was possible, and would be used to say "I neither confirm, nor deny that I ate the apples." as part of a conversational or interview strategy.
Your first sentence means that it was typical for me to eat a lot of apples. Your second one means (to me, an American) that it was not unusual for me to eat a lot of apples. In isolation here, both strongly suggest that it is no longer common for me to eat a lot of apples, but depending on context, that might not apply at all.