If you're using a modern browser, simply visiting the wrong website cannot infect your computer with a virus. However, downloading something and then running it, or installing the wrong addon/plugin can.
If the pop-ups were browser-based and only occurred when visiting the website, then they were probably just part of the website and there's nothing to worry about. However, if you're still getting them (or worse, getting them without even opening a browser) then it's likely that you have malware installed.
I'm a computer scientist and professional software engineer and use a PC full time at work and for at least 3-4 hours at home each day. I don't use any anti-virus software and yet I have never had any malware or virus on my PC: it is still as fast as the day I installed Windows on it 5 years ago.
The average family PC has anti-virus software, yet is festering with all sorts of viruses / malware / addware and bloat. My point is that being careful is infinitely more effective than relying on anti-virus software.
I will give you an analogy that sums up how effective anti-virus software is:
Imaging buying a pig that roams your house eating up undesirable things. Is it now okay to let every stray dog in the neighbourhood into your house to poop all over the place, because you know that the pig will eventually chase it out and eat up the poop? Definitely not!
The pig is like anti-virus software, and is about as effective at cleaning up viruses/malware as a pig is at chasing out stray dogs and eating up their poop.
My policy is it to let neither the pig nor the stray dogs into my home in the first place.