Why we are ticklish
Scientifically, being ticklish is an evolutionary tool for survival. No really. You get that special, slightly giggly feeling while a poisonous spider is crawling up your back. Now, with our fancy-dancy "buildings" and what not, we don't have spiders crawling all over our body too often. But once upon a time, we lived in the forest full of creepy crawlers that bite, and being ticklish was an advantage. Our most insecure body parts, such as our feet or armpits have more nerve endings than the rest of our bodies (therefore are more ticklish). The laughing doesn't laugh forever, and as a highly-ticklish person, I can admit that it gets old, quite quickly. After a length of time, the nerve endings send pain signals, and the tickling starts to hurt. People have been known to cry and/or throw up. Which means, if you're tickling someone, be responsible. If they ask you to stop, stop.
Some people are more ticklish than others. Some people aren't ticklish at all. The reasons for this are not solid science; nobody knows for sure. A popular theory is that it has to do with personality. If you are ticklish, while being tickled you can remain calm and try to avoid panicking. This usually dulls the tickling a bit.
Why we can't tickle ourselves
We cannot tickle ourselves. This is a good thing. If we could tickle ourselves, we'd be on our backs laughing every time we scratched our nose. Our brain, more specifically, the cerebellum distinguishes our expected and unexpected movements. With this, you body can know the differences between someone tapping you on the shoulder and you touching your own leg. If you're really that desperate to tickle yourself, some scientists with too much free time have created a machine that allows you to tickle yourself. The machine creates a short delay, and that delay is enough.
Sources:
http://ask.yahoo.com/ask/20010703.html
http://mfa.labradorstraits.net/faqs.html
http://www.howstuffworks.com/question511.htm