This syntax has a variety of names depending on what language you are using. The best way to find out what it is called is to look at the specification for the specific language.
HTML 4.x
I can't find any mention of this syntax in the HTML 4.x specification. It is not valid syntax.
HTML 5
In the HTML 5 specification the / character (called a SOLIDUS) is valid but has no effect for void elements such as <br />, <hr />, <img />, <input />, etc. and for foreign elements (such as SVG tags) it designates a start tag that is marked as self-closing. It is not a valid syntax for all other tags (such as <button /> mentioned in your question).
Then, if the element is one of the void elements, or if the element is a foreign element, then there may be a single U+002F SOLIDUS character (/). This character has no effect on void elements, but on foreign elements it marks the start tag as self-closing.
XML
According to the XML specification it is called an empty-element tag:
The representation of an empty element is either a start-tag immediately followed by an end-tag, or an empty-element tag.
XHTML
According to the XHTML specification it is called the minimized tag syntax for empty elements:
C.2. Empty Elements
Include a space before the trailing / and > of empty elements, e.g. <br />, <hr /> and <img src="karen.jpg" alt="Karen" />. Also, use the minimized tag syntax for empty elements, e.g. <br />, as the alternative syntax <br></br> allowed by XML gives uncertain results in many existing user agents.
C.3. Element Minimization and Empty Element Content
Given an empty instance of an element whose content model is not EMPTY (for example, an empty title or paragraph) do not use the minimized form (e.g. use <p> </p> and not <p />).
In general if you want to be precise I would recommend using the names as defined in the appropriate standard. Then if people aren't exactly sure what you mean they can look it up in the standard to find out. However if you don't want to use the name in the standard you are free to call it something else if you want. The important thing is that the people who communicate with you can understand you. I don't think anyone would misunderstand you if you used the term 'self-closing tag' for a tag in an XML document even if the standard officially calls it something else.
Thanks to Alohci for the HTML 5 reference.
