Do I have to pay again for a game I bought on a different PC with the same Steam account?

I recently bought a new laptop for gaming. I logged into Steam without difficulty, except when I logged into the account where I had bought my games on. My Friends list was empty, and I had no games. Am I able to play my games, on my laptop without re-purchasing them?

You do not need to re-purchase any game you've bought previously, as long as you log in using the same Steam account you previously bought the game with; however, you will need to download those games on the new computer.
It is possible that Steam will ask you to enter a code, sent to your e-mail address, when you attempt to log in using a new PC for the first time; this is a measure meant to protect accounts from being hacked and may be deactivated (not recommended).

My library is empty:

According to Steam Support, you are either logged into the wrong account, or your filter is not set to "All Games".

Please ensure that all games are visible by clicking View > All Games on the upper-left corner of the Library > Games list.

According to Steam Forum, it often happens that people accidentally create new accounts when trying to log into their old ones.

It appears that in some rare cases, programs may interfere with Steam.

How can I tell if the account I'm logged in with is my own?

• Click on [name]'s Account > Account Details located on the top right corner of Steam and check your Licenses + Subscriptions or your Store Transactions.
  You should see a list of all games you have purchased.

Should your library be empty, despite being logged in on the correct account, you may want to contact Steam Support directly.

Once you buy a game on steam, you can access it on any computer you own.

First make sure that you are actually on the same account that you bought the game in. If you hover your mouse pointer over the word LIBRARY at the top of the Steam interface, you should see a drop down. The "All Games" option will bring you to a list of all games you own, whether or not they are installed on your computer. Notably, this list does not include free games you do not have installed because owning a free game on Steam doesn't really mean anything.

If at this point there are games that should be on the list but are not, you should check the store page for a game you know you own. If it gives you a PLAY NOW button, then you can click that to install that game. Whether or not you see that button, you should contact Steam support at this point because you are experiencing completely abnormal behavior.