While it's certainly possible to put your website online and never touch it again, it's not a very good idea. A website should change over time by having new material added and even by altering the design or color scheme somewhat. A website that doesn't change becomes a dead website.
The emphasis of this article is on business and organizational websites, though some of the ideas can be implemented on personal websites as well. Here are my thoughts, in no specific order:
1. To keep existing customers happy, as well as adding new customers, it's important to keep things fresh. Add new pages with new articles or new products and information. Keep information up-to-date.
2. Remove products that aren't productive and delete or archive pages that are no longer needed. You don't want your viewers to have to wade through obsolete / discontinued products or outdated information. Links to any old pages that you want to keep available can be moved to an archived documents page.
3. Be sure there is an e-mail address or contact form available for customer feedback. Emails from (prospective) customers can help make you aware of possible changes or additions that need to be made to the site.
4. Changing the website's appearance or layout just a little every year, can help keep returning visitors interested. I'm never totally satisfied with the design of my websites anyway, so I like to tweak things every few months.
5. Make sure your links are all working properly. Nobody wants to click on a link that goes nowhere, or takes them to the wrong page or site. Websites come and go, so a link that may be good today might go nowhere tomorrow.
6. Add quality, reciprocal links. Reciprocal linking isn't really dead, so continue to (occassionally)
trade for good links from other sites. Don't overdo this. Adding one quality link per month is much better than adding 100 irrelevant, low-quality links.
7. Check your reciprocal backlinks to make sure the other site is still linking to you. Often links get moved or deleted. If your link is no longer on their site, you can either contact their webmaster and ask it to be restored or delete their link from your site.
8. Remove dead pages and images from your webhost's server. Removing the links to unneeded pages in your website is fine, but the pages still reside on the webserver and are just taking up space. Delete these pages and any images that aren't being used as well.
9. If you change the names of some of your pages or add or remove pages, be sure to update your google sitemap (if you have one) to reflect the changes.
In conclusion, keeping your website cleaned and updated will pay dividends in the long run.