How do people mine bitcoin?

Users usually use specially configured machines to mine bitcoin, as conventional computers are not meant to solve these technically challenging problems to mine bitcoin. Currently, it is estimated most of the bitcoin is mined in China due to the cheap power rates vs other parts of the world.

Why is there a price discrepency in bitcoin among the different exchanges?

For all the coins that have been mined, the price quotes are only from online exchanges, which are a small set of the total available number of coins available on the market. Another reason is prices in bitcoin is not updated instantly among the different bitcoin exchanges, and there is a time delay.

How do I get started with bitcoin?

You can start by going to an online bitcoin exchange, and making a purchase of bitcoin to see how the whole system/idea works.