Some sites generate some twisted alphabets or numerals and ask the user to retype?

I'm not exactly sure what you're asking, but I believe you're talking about CAPTCHA images. Are the "twisted alphabets" a picture of a bunch of numbers and letters that are distorted in a funky way? If so, then those are CAPTCHA images.

Say you're signing up for a new Yahoo! account. You provide your name and password, then you see a CAPTCHA image. This is used to ensure that it is a person, not a computer, that's filling out the form.

Sometimes people (usually spammers) write programs called bots that automatically sign up for a new account for them on a site, then use another bot to spam random people instead of doing it manually themselves. Those images help prevent this. So a bot will be able to insert a username and password into the form, but will not be able to get passed the CAPTCHA, since it is just an image with distorted numbers/letters the bot cannot read.

If this is not what you're talking about, then forgive me that I did not understand the question. If you'd like to learn more about these, you can look at http://en.wikipedia.org/wiki/CAPTCHA or just Google it.

Hope this helps.

They are verifying that you are not a web-bot, an automated program that tries to do those things that need this verification (like register as a user at a website). The web-bots can't (easily) read the images with the letters/numbers on them, humans can.

It's called a Turing Test. It's used to verify if the user is a computer.

Many spammer use computer automation e.g. to open multiple mail accounts on yahoo.

The original computer scrambles the original text in a way that is only human readable. However it is debated if these test even work as computer algorithms for this job, are getting better.