A lot of people expect owning a website to be like taking the bus, in terms of maintenance. They don't expect to do any work, and the maintenance gets offloaded to someone else — in the bus analogy, the transportation company; for your website, your hosting company.
However, owning a website is a lot more like owning a car under warranty. Some of the hosting and website maintenance is your responsibility, and you have to check it regularly to keep things working smoothly.
Here are some things our experts consider as important as changing your oil.
Everyone —no matter what
No matter who you are and what kind of hosting account you have, you've got to have a super-secure password that you change regularly. The Internet is a huge place and, unlike your car, someone can tamper with it from across the globe; they don't have to be a thug in your neighborhood.
This makes everyone susceptible to having their websites hacked and having malicious code inserted. "It won't happen to me" is the wrong attitude to have.
As the owner of a website, there's an implicit maxim that your information's up-to-date. If someone wants to get ahold of you, they should be able to using your website. You should update this information on your website as soon as it changes, lest you forget.
When you're no longer using your website, cancel it. Letting it fall into disrepair makes it susceptible to being hacked and causing grief for other Internet users. If you're no longer using it, either maintain it as vigilantly as you would if it were brand new and exciting, or get rid of it completely.
If you use an application for your website like WordPress or Joomla!, you need to check for patches very regularly. Once a week, every week, is not too often.
To be extra safe, updating your application's database password regularly is a good idea. However, this is slightly more complicated because you also have to update the password in the application. If you can figure out how to do it (it's really not that hard), you should.
Patches, patches, patches
If you're an admin worth your salt, you know this: you have to check for patches to everything installed on your server continually.
So you have some website maintenance. Not a ton, but enough to keep you safe, secure and online.