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The definable closure dcl(A) consists of elements definable from A with parameters. The algebraic closure acl(A) consists of elements satisfying a formula over A with finitely many solutions. In stable theories, dcl(A) ⊆ acl(A), and when dcl(A) is also algebraically closed it forms a submodel. These closures provide structure theory for models of stable theories.
From your work on definability, you know that a set S ⊆ Mn is definable over a parameter set A if there is a formula φ(x̄, ā) with ā ∈ A such that S = {x̄ : M ⊨ φ(x̄, ā)}. Now consider what elements are "pinned down" by parameters in A. The definable closure dcl(A) consists of all elements b such that the singleton {b} is A-definable — that is, φ(x, ā) has exactly one solution, namely b. Think of it as the set of elements you can uniquely name using formulas with parameters from A.
The algebraic closure acl(A) relaxes unique definability: b ∈ acl(A) if there exists some formula φ(x, ā) with ā ∈ A such that b satisfies φ and φ has only *finitely many* solutions. The element isn't necessarily uniquely pinned down, but it lives in a finite "orbit" over A. Notice the analogy with field theory: in the field ℝ considered as a structure, √2 is algebraic over ℚ because it satisfies x² − 2 = 0, which has exactly two solutions. Indeed, in algebraically closed fields, model-theoretic acl agrees with the classical algebraic closure from field theory.
The inclusion dcl(A) ⊆ acl(A) is immediate from the definitions: if b is the unique element satisfying φ(x, ā), then φ has finitely many (specifically, one) solution, so b ∈ acl(A). Both operations are closure operators: A ⊆ dcl(A), dcl(dcl(A)) = dcl(A), and similarly for acl. In stable theories, these closures behave especially well — acl(A) is always a model-theoretically small structure carrying the "algebraic content" of A, and independence (non-forking) is intimately tied to the acl operator.
The structural importance of these closures comes into focus when building models. A set A is algebraically closed (in the model-theoretic sense) if acl(A) = A — every element finitely definable from A is already in A. Algebraically closed sets serve as the "good" parameter sets for independence theory, analogous to how algebraically closed fields are the "good" fields in algebraic geometry. When A = dcl(A), A forms a closed substructure that reflects the ambient theory; combining both conditions gives the structural building blocks for understanding models of stable theories.