I understand that we share 1/2 of our DNA with a sibling or parent, 1/4 with our aunts/uncles, 1/8 with our cousins, etc. But I have also read that as humans, we share 98.4% of our DNA with chimpanzees. How can this be? How can I share 48.4% more DNA with a chimp than with my brother? Are there different “types” of DNA that I am not aware of?
The difference is that you’re comparing different things. 
Your relatives’ DNA is all still human, and the fractions you mention refer to variations inside that range (different blood types, for example). Of course there will be inheritance of identical genes from different sources as you go; your blood type could have come from your mother, your father, or both. The copies end up making your relatives more similar than the fractions would suggest.
Chimp DNA, on the other hand, includes some genes that aren’t human at all, and that’s where the odd couple of percent comes from. Most of the rest are essentially the same as human ones.
When they say you share 50% of your genetic make-up with a sibling, say, they mean that’s the percentage that is identical by direct descent from your parents. The actual percentage of similarity of the genomes will be much higher, I believe over 99% (I don’t recall the exact percentage for humans), because you derive the rest of it by descent from other humans, who are all related to some degree. Your mother and your father are related, you realize, although they may be twenty-fifth (or greater) cousins.
I never really understood how they did these comparisons (chimp vs. human or human vs. flatworm, etc.) - are they comparing genes or base pairs, for example? Are they including non-coding DNA too?
Thanks. That was basically what I was figuring - that the 98% number had to be - uh - questionable on some level.