If nothing precedes the turnstile $\vdash$, what follows is either an axiom or derivable entirely from the axioms. Basically, it means that you don't have to make any assumptions beyond the axioms to derive the following formula. $\vdash\text{(formula)}$ is equivalent to $\emptyset\vdash\text{(formula)}$
The “triangle union” in the Wikipedia article you linked in the comments is not a special notation. The $\Delta$ is just the Greek uppercase letter delta, and here the name of a set, and then $\Delta\cup\\{A\\}$ has then the obvious interpretation, the union of the set named $\Delta$ with the singleton set $\\{A\\}$.