What is the descriptive complexity of a set added by Cohen forcing?

Since Cohen forcing is weakly homogeneous, all hereditarily ordinal definable sets in the Cohen extension are already in the ground model. That applies in particular to any ordinal definable real, and (in even more particular) to any real with a lightface $\Sigma^m_n$ definition.

Cohen forcing (or any forcing adding a new real) adds new boldface sets on all levels. For example, if $c$ is the Cohen real, then $\{c\}$ is a new closed set.

Lightface sets may change. For example, the set of all nonconstructible reals has a $\Pi^1_2$ definition; this definition $\varphi(x)$ defines a set $A_\varphi=\{x: \varphi(x)\}$ in every model, but these sets are in general not equal; for example, $A_\varphi$ is empty in $L$, and nonempty everywhere else, e.g. in any Cohen extension.

(For a more trivial example, the set $\mathbb R $ changes whenever you add reals. The set $\mathbb R^{V[c]}$ is a "new" set in the Cohen extension. In fact, any perfect set will "change" in this way. This is probably not what you meant.)

Jensen proved in this paper that, beginning with $V = L$, it is possible to add a real $a$ by forcing such that $a$ is $\Delta^1_3$ in $L[a]$.

This result is the best possible, in the sense that one can never add $\Sigma^1_2$ or $\Pi^1_2$ reals by forcing. This follows from Shoenfield's Absoluteness Theorem (mentioned already by Asaf in the comments). In fact, Shoenfield's theorem implies that all $\Sigma^1_2$ and $\Pi^1_2$ reals are constructible; so they can't be added by forcing because they're already in the ground model. (For a proof, see Theorem 25.20 and Corollary 25.21 in Jech's book).

I do not know whether anyone has improved on Jensen's result to show that a $\Delta^1_3$ real can be added to any ground model by forcing.

Jensen's forcing is a bit difficult to understand. For an easier-to-understand example of a non-constructible $\Delta^1_3$ real, there is $0^\sharp$. Even if $0^\sharp$ does exist (which is not provable in ZFC -- it is a large cardinal axiom), it cannot be added by any set-sized notion of forcing. So $0^\sharp$ does not answer your question, but it seems related, so I thought I'd share.