Yes you can grow them. I'll leave it up to you to decide whether you want to or not. Most apple trees grown from seed do not turn out anything like the juicy crisp apple you just ate. Because of cross polination with other types you just really don't know what you'll get. Nothing wrong with that BTW. Trees grown from seed will also be a "standard" size tree which will be very large and it takes YEARS of proper care for them to make apples Most apples are propagated by grafting bud chips onto a dwarfing rootstock so the trees stay a manageable size. But,hey, if you want to experiment and are not looking to grow that red or golden delicious like you just ate, I say go for it. All of the varieties we have now were grown from seed at some point and cross pollination is why we have so many different types. Just be aware that it could be crossed with cooking apples or yellow or red anything really and it probably won't produce fruit like what you had. But then again it might.
I would start it in a pot first and then transplant it this fall ( or you can bury the pot and transplant next spring) Then you need to do some research on training apple trees. It's first year it will make a "maiden whip" and you have to trim it a certain way to get that familiar "apple tree" shape. Or if you have plenty of space, just let it grow! Good luck!
This site tells you how to prepare and propigate apple seeds. Stratification is required. This article also talks some about what jlmann was talking about as far as not knowing what you will get. Many commercial apples are pollinated by crabapples so you really don't know the genetics of what you have if you go with an apple from the supermarket.
I planted crabapples in my orchard as I have one early tree I'm not confident I have a pollinator for. I'm hoping the crabapples will bloom early enough to take care of that problem. I planted them this year so I won't know until next year.