There are many different ways of having your own site to display your photos etc and a lot depends on how much time and probably money you want to spend on it.
There are a lot of web based galleries that you can get, but they all look similar and I don't think you can customise them very well.
You could get a bit of webspace, a lot of companies offer it free, say 30mb or so (most ISP
's offer a bit). Get a HTML editor and create a few pages with your pictures on. If you have a lot of pics though it could take quite a while to maintain.
Or you could spend a bit of money on a webspace package and a lot of time learning a scripting language like php and create a photo gallery exactly how you want it. There are many tutorials on the net and once you get going you can create galleries etc and make them look exactly how you want them to, but it takes a lot of time to learn.
My site if your interested http://www.plane-mad.com