Step 1. Create a Bitcoin wallet.
A bitcoin wallet is website or program which you can use to hold store bitcoins and make transactions. This website offers a free wallet service, sign up takes a few seconds and all you need is a password. Simply click the start a new wallet button at the top of the page. Alternatively you can use the official bitcoin.org desktop client however some setup time is involved.
Step 2. Locate Your Bitcoin Address.
Once you have created a bitcoin wallet login and click receive money or receive coins. On this page will be a "bitcoin address" which is 34 - 36 character long and starts with 1. This address is all you need to receive payments.
Step 3. Buy With Cash.
Go to the Blockchain.info homepage and enter your bitcoin address from Step 2. into the search box. This will take you to a page which will show all transactions made to and from your new address, however as this address is new nothing will be displayed yet.
Alternative Methods
Bitcoin Exchange - A bitcoin exchange is a place we people buy and sell bitcoins for other currencies. We recommend Mt.Gox and Intersango. Once you have signed up for an account you will be able to wire money directly into the exchanges bank account. You then need to create an "Ask" or "Buy" order and buy the appropriate amount of bitcoin (BTC) at the current exchange rate. When you have completed a trade withdraw to the bitcoins to the address from step 2.
Free Sites - You can get a small amount of free bitcoins each day from dailybitcoins.org