How can I update my OS X 10.5.8? This is the first time I've ever had a computer that didn't play out on me before it needed an upgrade and now I'm lost. This is also my first Mac laptop, and I am running OS X 10.5.8 (Yes, I realize it's old…I had it put up for a while)

Can someone tell me the next step or steps that I can take to do an upgrade?  I have no clue which animal I'm running or what I need to go to. I'm noticing more and more sites/actions are telling me that they are not compatible because my system is no longer up to date.
See here: http://www.apple.com/osx/how-to-upgrade/

TL;DR: buy a DVD to update to Snow Leopard and then update it to Mavericks (if your hardware supports it).