When a person makes eggs or sperm, half his or her genes get parcelled out in fairly random fashion into each gamete. And every child gets half his genes from his mom, and half from his dad.
So, you share 50% of your genes with each of your parents, right?
Well, probably. It depends on whether or not your parents have genes in common; if they're from the same ethnic group (for instance, if they're both blonde, blue-eyed Swedes), they almost certainly have some DNA in common. If a tribe of people settles in an isolated valley, after a few generations everyone in that tribe is going to be related to everyone else, even if everyone's careful not to accidentally marry a first cousin or uncle. If your parents have some genetic relation to each other, then you could have more than half your genes in common with one or both of your parents.
But, for the purposes of this writeup, we'll go with the 50% figure.
Now, let's talk about your brothers and sisters. Full brothers and sisters share the same parents, and because of the random dispensing of genes, it's generally thought that siblings share about 50% of their genes with each other.
Things, of course, aren't that simple. If your mom is unusually homozygous then she will be passing out pretty much the same genetic set to you and your siblings. So no matter how varied your father's genes are, in that case you and your sibs would have somewhat more than half your genes in common. If both your parents are largely homozygous, then you and your sibs would have lots of genes in common, even if your parents are from very different ethnic groups.
Conversely, it's possible through the magic of random assortment for two siblings to get completely different genetic halves from each parent. And to complicate matters further, there's always the chance of mutation to give you genes that none of your relatives have.
With the previous caveats in mind, here's the breakdown of the percentage of genes you likely share with various relatives (with possible deviance in parenthesis):
As you can see, unless your ancestors have significantly shared genes, past your second cousins there's not much blood relation there at all.
So, if your grandfather really freaks you out, you can sleep safe in the knowledge that you really might not share any of his genes at all.