Could you help me understand the difference between "OTHER" and "ANOTHER"

 I would say another means additional. You are adding something on.

Let's say we have a mother and a daughter. The mother gives her daughter one cookie. The daughter wants two cookies, so the daughter asks, "Can I have another cookie?"

Other: you would use as if you want something else. Other means you have a choice and are being specific about it.

The mother picks up the cookie for her daughter. Her daughter says "I don't want THAT cookie. I want the OTHER one."

Another means that you would like one more of something, that you have already had.  One more of the same.  Significa que quieres algo más, algo más de lo que ya has tenido.  Please give me another glass of water; I am very thirsty.

Other means you would like something different.  Significa que quieres algo differente de lo que has tenido.  Please give me an other glass of water; this one is dirty.   Necesitas poner "an" en frente de "other" porque vienen dos vocales juntas.

Espero que eso ayuda,

A

another is usually "singular" while other is plural.

i.e.:  May I have another bottle of water?

Other students want to go on the class trip as well.

 Other es un pronombre definido mientras que another es un pronombre indefinido...

Por ejemplo....

Dame otro lápiz (cualquiera) --> give me another pencil

Dame el otro lápiz  --> give me the other pencil

(Un ejemplo bastante malo, por cierto...)