Difference between 这儿 and 这里？
The former is a lot more casual and easier to say, although when spoken, most people just say 这
I think this explanation has a dialect bias. The people I know who speak Chinese natively are from southern parts of China and never say 这儿. I have never asked them but I don't think they would agree that it is easier to say. (Personally I find 这里 to be more convenient. I guess it depends on whether you use a lot of the 儿 sound in general or not. Things you say often will probably feel easier. From a learner's perspective this depends on what languages you already know. For speakers of many US English dialects 儿 will probably fit nicely in the mouth.)
The difference could be summarized by an anecdote from Kunming (in southern China) where I once ended up in a conversation with people from the US and Kunming, China. When the native English speakers were talking to each other someone mentioned that it sounded like they came from Beijing, because of the thick 儿 sounds in their English dialect. In the north people use 儿 at the end of many words, not only 这儿. The only word I regularly hear with an 儿 is 馅儿. And the best 馅儿 is put in a dumpling, a northern dish. And sometimes, I think, 眼镜儿， which makes it easier to distinguish from 眼睛。
“这里” is correct, it can be used in writing and speaking. "这儿", "儿" is pronunciation habit from Beijing dialect, most of time, northerners would use it in speaking. "这" or "这呢", most of time, southerners would use it in speaking.