A few days ago I took apart my laptop to see if my processor was removable. My dad was watching over me and after I removed the heatsink (I think that's what it's called. It's the part connected to the fan and DIRECTLY above the processor) a small greasy sticker-like thing ripped, half was left on the processor and the other half the bottom of the heatsink. He said that it was processor grease needed to keep my PC from overheating and that I should replace it.

Is this true, or was it something else like a "void warranty if broken seal"?Sounds like it was a thermal pad, used to transfer heat.
You need to replace this with some thermal paste after cleaning the remainder of this pad, or your CPU will overheat. Do not run that laptop without properly applying new thermal paste or you may cause irreparable damage.
Your warranty may be void now that you've ripped the pad though, it depends on the terms and conditions.most warranties are void the moment you open the case.That all depends on the terms.
The warranty on my laptop was not voided by me opening it up, adding more RAM, and replacing all the thermal pads with proper thermal paste.
There were also no "warranty void if..." stickers. On other computers however those stickers are everywhere. I think it depends what mood the guy on the assembly line is in Thanks everyone I'll replace the thermal paste right AWAY

Today I went to frys and bought a "non silicone heat transfer compound". Is this the same thing?
Pics:



Yep, that's thermal paste.
Not high quality paste, but it'll do the JOB, and there's usually not a massive difference anyway between different pastes.
If it didn't come with instructions, all you need to do is FIRST clean the processor and heatsink, with a soft wipe or tissue paper and a dab of rubbing alcohol, until all the old paste is gone. Then, wait for it to dry out (don't touch either the heatsink or CPU with your finger) and apply a small amount of this compound to the CPU - about the size of a ball bearing, maybe a little less. Get the heatsink back on, fan connected etc, and you're ready to rock.
Good luck.Thermal Paste Guide