Soundness means that you cannot prove anything that's wrong.
Completeness means that you can prove anything that's right.
In both cases, we are talking about a some fixed system of rules for proof (the one used to define the relation $\vdash$ ).
In more detail: Think of $\Sigma$ as a set of hypotheses, and $\Phi$ as a statement we are trying to prove. When we say $\Sigma \models \Phi$, we are saying that $\Sigma$ logically implies $\Phi$, i.e., in every circumstance in which $\Sigma$ is true, then $\Phi$ is true. Informally, $\Phi$ is "right" given $\Sigma$.
When we say $\Sigma \vdash \Phi$, on the other hand, we must have some set of rules of proof (sometimes called "inference rules") in mind. Usually these rules have the form, "if you start with some particular statements, then you can derive these other statements". If you can derive $\Phi$ starting from $\Sigma$, then we say that $\Sigma \vdash \Phi$, or that $\Phi$ is provable from $\Sigma$.
We are thinking of a proof as something used to convince others, so it's important that the rules for $\vdash$ are mechanical enough so that another person or a computer can check a purported proof (this is different from saying that the other person/computer could create the proof, which we do not require).
Soundness states: $\Sigma \vdash \Phi$ implies $\Sigma \models \Phi$. If you can prove $\Phi$ from $\Sigma$, then $\Phi$ is true given $\Sigma$. Put differently, if $\Phi$ is not true (given $\Sigma$), then you can't prove $\Phi$ from $\Sigma$. Informally: "You can't prove anything that's wrong."
Completeness states: $\Sigma \models \Phi$ imples $\Sigma \vdash \Phi$. If $\Phi$ is true given $\Sigma$, then you can prove $\Phi$ from $\Sigma$. Informally: "You can prove anything that's right."
Ideally, a proof system is both sound and complete.