って is a casual variation of the
と particle that is used for quoting something. This use is called 引用（いんよう） in Japanese, which basically translates to ‘reference’, or ‘citation’.
って is almost exclusively used after direct quotations, and will be followed by a verb indicating what type of communication was used. 言（い）った, 思（おも）った, 聞（き）いた, for example.
-
「行（い）ってきます」ってママが言（い）った。
Mom said, ‘I’ll be back soon’.
-
「それは危（あぶ）ないよ」って先生（せんせい）が言（い）った。
The teacher said, ‘That is dangerous’.
Just like with
と, the verb that comes after
って is frequently omitted. This is even more true for
って, as many things are omitted in casual language.
Although often heard, due to って being very casual, it can be considered unnatural to use in sentences that also use the です or ます polite structures. However, if the conversation is formal but friendly (like between a senpai and a kouhai), most people would consider it normal.