AFAIK A ⊢ B means that B can be deduced/derived from A, or there is a proof of B from A.
And M ⊨ N means M is a model for N , i.e., M is an interpretation in which all wffs in N are mapped into truths. Look up the meaning of interpretation.
The first '⊢' is syntactic, dealing with provability, so that B can be deduced/proved from A, i.e., there is a finite collection of sentences starting with A, where each is either a theorem, or follows from previous sentences by the application of some inference rule(s), the last sentence of which is B. 'A' on the left may be the empty set, in which case B is a tautology.
This deals only with formal relations between sentences, and not with their actual inner content.
The second , i.e., '⊨' deals with semantics, or the notion of truth. Informally, M is a possible world in which the wffs are all feasible/realizable. This deals with truth.