Programming language
Each one has to define what it means by that. But overall I would say it is to create a new syntax, usually through metaprogramming.
The mechanisms can be simpler, from the preprocessor of the C, until very sophisticated macros who may even involve the compiler.
It is very common that these languages are appropriate for Dsls. A example.
In general this is very problematic and usually pee in the hand of those who are not very good at dealing with all complexity. Programming language explodes into possibilities of combinations. Since "small" I wanted to make a language like this, today I know how it is almost impossible for a product that goes beyond a very specific niche or academic research.
Some languages force the bar saying that they are extensible because you can extend something, but little.
Wikipedia.
Declarative language
There are declarative languages that are already Dsls that allow the creation of new statements in a simple way. Although some of them are extensible by definition, I saw nothing official indicating that they are considered "extensible languages", so despite the term fit, it is used in a different context.