Updating a server
You can also copy and replace the server files, but there is a gui available as shown below.
Download the server jar file
Visit the corresponding site for your server type (as displayed in the table below) and download the server jar file that you want to update your server to. Make sure that you update to the correct server type (spigot -> spigot) and not (spigot -> taco).
You can find a list of downloads for common server types here.
No API jars
Make sure that you don't download an api jar! They can be identified by the '-api' tag in their name.
Select the server jar file
Now it's time to select the server jar file you downloaded in the previous step. You can either drop it onto the form or you can browse for it with the 'Select file' option.
Start the update
Click 'Update' to start the process. This won't take long.
Wrapping it up
All done! Wasn't that easy?