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In a model M, the definable closure dcl(A) is the set of elements definable by formulas with parameters from A; the algebraic closure acl(A) is the set of elements in finitely-defined sets from A. These notions generalize field-theoretic closures and provide a dimension notion for any model. Independence of sets is captured via forking: sets are independent if no element in one is algebraic over the other.
You already know from definable and algebraic closure that a model M provides two natural ways to extend a parameter set A inward. The definable closure dcl(A) collects every element b in M that is uniquely pinned down by some first-order formula with parameters from A — that is, the formula φ(x, ā) is satisfied by b and b alone. The algebraic closure acl(A) is more permissive: it collects every element b that lives in a finite set definable with parameters from A. The formula φ(x, ā) may have finitely many solutions, and b is one of them. So dcl(A) ⊆ acl(A) always. In a dense linear order, acl(A) = A itself because no finite set of points is definable from finitely many endpoints unless the point is already there. In an algebraically closed field, acl(A) is exactly the field-theoretic algebraic closure of A.
The field analogy is the right one to carry forward. In field theory, a set of elements is algebraically independent over a base field F if no element of the set is algebraic over F and the others — that is, no element satisfies a nonzero polynomial with coefficients in F ∪ {the others}. Model theory generalizes this: a set B is independent over A if no element of B is algebraic over A ∪ (B minus that element). The precise technical definition uses forking: B is independent from C over A if the type of B over A ∪ C does not fork over A. In stable theories, forking captures exactly the failure of algebraic independence — an element forks over A if it is algebraically constrained by the new parameters beyond what A already forced.
The payoff is a robust dimension theory that works for any sufficiently well-behaved model. A basis of M over A is a maximal independent set; all bases have the same cardinality, called the dimension (or Morley rank in the categorical case). This mirrors the fact that all bases of a vector space have the same cardinality, or all transcendence bases of a field extension have the same degree. The model-theoretic version applies not just to fields and vector spaces but to any stable theory — including certain theories of graphs, groups, and combinatorial structures — wherever forking defines a well-behaved independence relation. Recognizing that dcl, acl, and independence are three interlocking tools that together give a model-theoretic substitute for linear algebra is the central conceptual step in understanding stability theory and its applications.