The process than is simply putting your CD in your computer CD drive and then running the mp3 ripping software. Each software program has its own way of letting you do this and you will need to become familiar with each program. I recommend Easy CD Ripper as one of the easiest rippers to get started with and use.
Although this method is popular, there are some risks. Most music is copyrighted and if you don't own the CD of the songs you want to acquire, you will technically be breaking the law. Also, there is the risks that downloading files over the Internet will contain viruses or other unwanted material although the majority of files will probably be safe.
Other ways of getting MP3s is from free MP3 sites like EMusic or by copying files from another person's MP3 format CD that they have created.