Keeping Your Website Going
Websites need care & attention if they are to continue providing value to you (the website owner). Just like owning a car, a house, a computer - if you don't maintain it, gradually things degrade. Small problems become big problems. And sooner or later there will be a failure, which of course is bound to happen at the worst possible time...
So the question is, what needs doing and why? (NB. the "how" very much depends on what you're website has been built with, and can vary hugely).
1. The 'Content Management System'
It is a rare website these days that is not powered by a content management system (CMS) of some sort. WordPress, Drupal, Joomla, Wix, Weebly, Squarespace, Shopify and on and on. They are all basically a software package that assists you in creating and managing a website without having to do coding or computer programming. Some of them allow you to run your own independent system (WordPress, Joomla, Drupal), some of them are large umbrella systems managed on your behalf (Wix, Weebly, Shopify, Squarespace).
So you need to know if you are responsible for keeping this underlying software package up-to-date, or if someone else is. If it's someone else, all good - just make sure they're are doing what they said they would. If it's your job - make sure you are checking for updates on at least a weekly basis. The content management system is like the engine for your website, it powers it and drives it and makes it work. A stuttery engine will give you stuttery website performance.
Why update the core CMS? Because there are frequent fixes and updates that address security issues (people breaking into or abusing your site), bugs (things that don't work properly) and improvements (making things work better and more smoothly).

2. The Add-Ons & Extensions
You are quite unlikely to be using the base-level CMS. For a website to be useful to you and what you want to do with it, you are going to add extra functions to the site - called 'add-ons' or 'extensions', or maybe 'options' in the big shared systems. For an independent website, these add-ons also need the same upkeep as the underlying website system. Again, is that your job or is someone else doing it? You need to know or you could wake up one morning to a day that is going to be wasted on fixing a broken website (not to mention the potential loss of business while it's down, and the cost of repair).
Why update the add-ons? Because there are fixes and updates that address security issues (people breaking into or abusing your site), bugs (things that don't work properly) and improvements (making things work better and more smoothly).
3. The Webhosting Account
Often overlooked, the webhosting is where the website 'lives'. Think of it as renting office space in an office block - there are multiple businesses not connected to you all sharing the same office block, and you have rented one corner of one floor. Your rented space is your own webhosting account, and the other businesses are other independent websites.
What if the owner of the office block never did any maintenance? And worse, you can't do it yourself even if you wanted to because your contract forbids it? Now you've got a beautiful office in a shabby building with poor security. Not good.
So it's important to know if the webserver that manages your webhosting is being maintained and managed for you. Once more, there are always ongoing changes, fixes, updates to the webserver software - and you are nearly always not allowed to do anything to the webserver (because you would affect all the other inhabitants too). Ask the question of your webhosting provider - "do you managed and maintain the webserver, and keep it up-to-date?"
Why update the webserver? Because there are frequent fixes and updates that address security issues (people breaking into or abusing the entire system), bugs (things that don't work properly) and improvements (making things work better and more smoothly).