Two systems on one iMac - how do I delete one?

Our last Mac died and we bought an iMac early last year which came with Lion/Mountain Lion already installed. I then installed Mavericks this year.

I had backed up our old mac and managed to transfer the files to the new iMac but I seem to have both Snow Leopard (backed up files) and Mountain Lion/Mavericks running independently. I can log out of one and then log into the other which I did to ensure all - or at least - most of my files were available in Mountain Lion.

I have all the files I need and the computer just seems to be running so slowly so is it possible to now delete Snow Leopard? Will it make my mac any faster and will I lose everything I already have?

I am sorry to be such a fool but I really don't understand how computers work and my husband is much better at woodwork!

Please can someone help me?

Thank you.

Corky's mum!