Time Machine - problems between tiger/leopard/lion?

Hello, I am thinking of selling my 2007 Macbook (upgraded with 2 gb vram but replaced hard drive many times). I want to buy a new Mac. Will I be able to use my files that are backed up on Time Machine (10.6) on my new computer (10.8)? I ask because, when my hard drive died 10 months ago, my dad attempted to replicate it in order to send it to me (I live in Thailand, but my back up is in America. Smart, I know.) He is a PC person, but took it to several computer people who could not figure out how to replicate it. They said it would take a program costing big bucks (perhaps they were taking advantage of my dad's naivete). Long story short I decided to wait to deal with myself.

Is time machine some how connected to a specific computer or can it be applied to any computer? In other words, can I sell my computer now, or should I wait until I use it to get the files from Time Machine?