Not sure why people have had difficulty with this but the majority of problems above are down to your browser cookies - they remember your log-in.
You can delete your cookies but that’s a little drastic.
A very easy way is to access one account on 1 browser and another account on another browser eg. Internet Explorer, Firefox, Safari, Chrome, etc. Each browser will save its own cookies.
The best way however is to use the twitter app on your phone which provides the ability to switch from one account to another.
If you wish to use just one browser, and prefer to set it up from that browser, just log-out and then create a new account from the home page. Do not login, as that will take you to your original account, but you will need to set up the additional account(s) with different email addresses.