Why is my memory full when i have only used half of it?

Hi all, i have a 2012 MBA 120GB, i am new to mac books but have had all the other apple products for a long time.
my MBA is constantly telling me "memory full" and other messages like that. i have looked at "about this mac" and it displays i have 60gb of music, photos, apps etc, and the other 60gb is "other", and therer is no memory left.
i have deleted all of my emails in all boxes to see if this is it, and it isnt. i dont have any back ups, for our iphones and itunes at all, and i havent set up time machine either. i am at a total loss as to why this could be.
i have avoided using it latley because i cant put anymore photos or music on to it. the only thing that i have done to it since i have had it is install Mountain lion, and microsoft office for mac.
i have looked around for the answer to this and have come up with nothing, i'm no expert at all with a mac, i have looked at installing daisydisk or something like that, but i dont have enough memory to install it!! so if anyone has a step by step guide as to how to sort this out i would be REALLY grateful!!