Mark up means to add annotations (the act of adding highlights, notes, or markings).

For example, take this plain text:

This is a title
This is a subtitle

In HTML I would annotate (mark up) this text using HTML tags:

<h1>This is a title</h1>
<p>This is a subtitle</p>

The browser reads these tags and displays the content between the tags appropriately:

This is a title

This is a subtitle