I guess you use the symbol $\vdash$ to mean the derivability in a particular deduction system $\mathcal{D}$ (there are many deductive systems), so the proper notation would be $\vdash_\mathcal{D}$.
Often, when no ambiguity arises, the subscript $\mathcal{D}$ is omitted because many deductive systems are equivalent and you are not interested in the specific syntactic definition of the deductive system $\mathcal{D}$.
Writing $\vdash_\mathcal{D} A$ means that the formula $A$ is derivable in $\mathcal{D}$ without any hypotheses, i.e. without any assumption other than the logical axioms of $\mathcal{D}$. It does not mean that $A$ is necessarily a logical axiom of $\mathcal{D}$, because from logical axioms other formulas can be derived by means of the inference rules of $\mathcal{D}$.
To explain better this, it is necessary to be a little more precise.
The deductive system $\mathcal{D}$ is made up of logical axioms (some tautologies that serve as premises or starting points for further reasoning) and inference rules (that allows you to derive a formula from other formulas).
A derivation in $\mathcal{D}$ from the hypotheses $B_1, \dots, B_m$ to the conclusion $A$ is a finite sequence of formulas $(A_1, \dots, A_n)$ such that $A_n = A$ and, for all $1 \leq i \leq n$:
- either $A_i$ is an hypothesis (i.e. $A_i = B_j$ for some $1 \leq j \leq m$);
- or $A_i$ is a logical axiom of $\mathcal{D}$;
- or $A_i$ is obtained by applying an inference rule of $\mathcal{D}$ from the premises $A_{i_1}, \dots, A_{i_k}$ (where $i_1, \dots, i_k < i$).
We write $B_1, \dots, B_m \vdash_\mathcal{D} A$ if there is a derivation in $\mathcal{D}$ from the hypotheses $B_1, \dots, B_m$ to the conclusion $A$. In particular, we write $\vdash_\mathcal{D} A$ in the case there is a derivation with no hypotheses, i.e. the derivation is obtained by applying only the cases 2 and 3 above: $A$ is either an axiom of $\mathcal{D}$ or obtained from the axioms of $\mathcal{D}$ by applying the inference rules of $\mathcal{D}$.
For instance, suppose that $A$ is an axiom of $\mathcal{D}$ (hence $\vdash_\mathcal{D} A$) and a tautology, so $A \lor A$ is still a tautology but (possibly) $A \lor A$ is not an axiom of $\mathcal{D}$. However, $A \lor A$ can be derived from $A$ using the inference rules of $\mathcal{D}$, hence still $\vdash_\mathcal{D} A \lor A$.
The hypotheses $B_1, \dots, B_m$ can be seen as non-logical axioms, i.e. formulas that are not tautologies but that you assume to investigate their consequences. They can be the axioms of a specific mathematical theory, for instance Peano arithmetic or group theory.
Which are the logical axioms and the inference rules of $\mathcal{D}$? It depends on the deductive system $\mathcal{D}$. Some deductive systems (such as natural deduction) are essentially without logical axioms, but even in that case it makes sense to write $\vdash_\mathcal{D} A$. Indeed, such deduction systems have an inference rule (sometimes called deduction theorem) that allows hypotheses to be discharged, i.e. such that if $B \vdash_\mathcal{D} A$ then $\vdash_\mathcal{D} B \to A$.
To give a concrete example, let $A$ be any formula. How can we prove the formula $A \to A$ (which is a tautology) without any hypotheses? Clearly, $A \vdash_\mathcal{D} A$ (case 1 in the definition of derivation: if you suppose $A$ then you can conclude $A$) an then $\vdash_\mathcal{D} A \to A$ by the deduction theorem.
