"This is a blue coat."
これ is used when "this" takes the place of a noun. It basically means "this thing." この is used when "this" is used to specify something, i.e. "this coat."
In the above sentence, これ is acting as the topic, hence the use of the は particle. It's because of this that in the English translation, "this" is acting as the sentence subject. You are stating that これ/this is a blue coat.
I don't know exactly what sentence you had before this, but to make something using この you might say このコートは青いです which means "this coat is blue." In a case like this, この is used instead of これ because you are using it to specify which coat. "This" coat as opposed to "that" coat. The は particle comes after このコート in this case because this is the topic, and the purpose of the sentence is to say that it's blue. "This coat" becomes the subject in the English translation in this case.
Because then it would be "this coat is blue," which has a slightly different meaning (emphasizing that blue is the color of this coat, rather than the fact that this item is a blue coat). That translation can't be accepted because it is necessary to differentiate between これ and この in these cases.