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A sequence of functionals (fₙ) in X* converges weak* to f if fₙ(x) → f(x) for every x ∈ X. The weak* topology on X* enables compactness: the closed unit ball is weak* compact (Alaoglu's theorem).
You now know two modes of convergence for sequences in a normed space: norm convergence (strong convergence), and weak convergence, where xₙ → x weakly if f(xₙ) → f(x) for every bounded linear functional f. Weak* convergence lives one level higher: instead of testing vectors against functionals, it tests functionals against vectors. A sequence (fₙ) in the dual space X* converges weak* to f if fₙ(x) → f(x) for every fixed x ∈ X. The "star" marks that the dual space X* is now the space being tested, and the original space X provides the test functions.
The distinction between weak and weak* convergence matters when X is not reflexive. Recall from your dual spaces prerequisite that the dual of X is X*, and the double dual is X. Weak convergence in X* means testing against elements of (X*)* = X — every functional on X*. Weak* convergence tests only against the elements of X sitting inside X via the canonical embedding. This is a strictly coarser topology when X is not reflexive: there are more weak* convergent sequences than weakly convergent ones in X*. In a reflexive space, X = X and the two topologies coincide.
Why does this weaker topology matter? Because it enables compactness. The Banach-Alaoglu theorem states that the closed unit ball in X* is compact in the weak* topology — for any normed space X. This is a profound statement because the unit ball in an infinite-dimensional space is never compact in the norm topology (Riesz's theorem). Weak* compactness rescues compactness arguments that would otherwise fail in infinite dimensions, and is the engine behind many existence proofs in analysis, PDEs, and optimization: extract a bounded sequence of approximate solutions, use Alaoglu to find a weak* convergent subnet or subsequence, then show the limit is an exact solution.
A concrete instance: in L^∞([0,1]), the dual of L¹, a bounded sequence of functions (gₙ) in L^∞ with ‖gₙ‖∞ ≤ 1 always has a weak* convergent subnet. Weak* convergence here means ∫gₙh → ∫gh for every h ∈ L¹. This limit g need not be pointwise limit, it need not converge in norm, but it exists and is bounded — which is enough for many applications. This "weak* limit extraction" technique appears throughout harmonic analysis, probability theory (as tightness and vague convergence of measures), and the calculus of variations.