In Thunderbird you can tag a message with one or more tags by clicking on the message with the right mouse button and choosing Tag. There are five standard tags with descriptions Important, Work, etc. If the mail server is an imap server like post.science.ru.nl, then the tags are stored on the server, which means that every Thunderbird will show the tags.
It is somewhat more complicated for new tags that you can create. Thse are also stored on the imap server, but will initialy not be visible in other Thunderbird programs. One can make a new tag by clicking with the right mouse button on a message and choosing Tag -> New Tag... or using the menu Tools -> Options... -> Display -> Tags -> Add. The tag name will be used both as internal tag name and as visible description. If you alter the tag name later on, then the internal tag name will remain the same. The internal tag name (i.e. the initial tag name) will be used as tag on the imap server. A new tag will only be visible in another Thunderbird if a tag with the same internal tag name exists. So it is important the choose the right initial tag name. In particular one has to agree on a tag name if the mailbox is shared. After that everyone can change the colour and tag name, without consequences for the visibility of the tags.
The difference between the internal tag name and the visible tag name is goed te zien in the Config Editor (Tools -> Options... -> Advanced -> General -> Config Editor... and then filter on the word tag). Zo zie je onder meer dat het internal tag name Belangrijk tag $label1 is.