After the phrase in Russian, "Y menia..." (I have...) why is there normally an "ect?" However, sometimes there is not this short little interjection. What purpose does it serve and when do I use it?
It is a form of "to be" that states existence. "By me, there is a ..." would be a literal translation to English, though you do not speak like that in English.
When the existence is not questioned ("I have long legs") or not the point of the sentence ("We have many students") the word «есть» is not used. It does not make much sense to proclaim the existence of something if what you really wanted to say was to describe ots properties or quantity.