While あります and います are used to say "I have" or "there is" for inanimate and animate objects respectively, もっています and もっている are the formal and informal present tense conjugations of the verb もつ - "to hold/to carry", "to possess". That does mean, of course, that あります and もっています can sometimes be used interchangeably, but that's all down to the circumstances.

あります is usually used for inanimate objects. います is for used for animate objects (like people, animals, etc). もっています and もっている are exactly the same verb and used with the same nouns, but those are two different verb conjugations. もっています (ます form) is more formal もっている is usually called by the `dictionary form`, and it is less formal.

"To have" has many translations in Japanese and other verbs and words, too, so we have to choose one of them when we translate sentences. There are cases that it's better remembering whole the sentence with translation. Examples: います--> I have a younger sister. わたしにはいもうとがいます。 います--> Two boys are in the classroom / There are two boys in the classroom. きょうしつにはふたりのおとこのこがいます。 あります--> A book on the desk / There is a book on the desk. つくえのうえにほんがあります。 もっています/もっている--> I have a cellphone. (formal form) わたしはけいたいでんわをもっています。 (informal form) わたしはけいたいでんわをもっている。 *formal form is made with です and ます. other common translations: I had lunch with my friend. わたしはともだちとおひるごはんをたべました。 I had a coffee, too. わたしはコーヒーものみました。 I have a cat. わたしはねこをいっぴきかっています。 I hope this helps you.