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A vector field F is conservative if F = ∇f for some potential function f. Conservative fields have zero curl (∇ × F = 0 for continuous partials) and satisfy the property that ∮_C F · dr = 0 for any closed curve C. Magnetic fields are irrotational models of conservation.
The Fundamental Theorem for Line Integrals — your prerequisite — says that if F = ∇f, then ∫_C F · dr = f(B) − f(A), where A and B are the endpoints of C. This is the multivariable analogue of the Fundamental Theorem of Calculus: the line integral depends only on the values of f at the endpoints, not on the path taken. A conservative vector field is precisely one for which this path-independence holds. The name "conservative" comes from physics: in a conservative force field, the work done moving a particle depends only on start and end position, so energy is conserved (no energy is gained or lost by taking a roundabout path).
The central equivalence theorem (in a simply-connected domain) is: F is conservative ↔ F = ∇f for some scalar potential function f ↔ ∮_C F · dr = 0 for every closed curve C ↔ the curl of F is zero (∇ × F = 0). Each of these four conditions implies all the others. Zero curl is the easiest to check computationally — it only requires partial derivatives. For F = ⟨P, Q, R⟩, the condition is ∂P/∂y = ∂Q/∂x, ∂P/∂z = ∂R/∂x, ∂Q/∂z = ∂R/∂y. In R² this reduces to ∂P/∂y = ∂Q/∂x.
When you have confirmed F is conservative and want to find the potential function f, the method is systematic: since F = ∇f means ∂f/∂x = P, ∂f/∂y = Q, ∂f/∂z = R, integrate the first component with respect to x (introducing a function of y and z as the "constant"), then differentiate with respect to y and match against Q to pin down that function, then differentiate with respect to z to pin down any remaining constant. Each integration step is like running the Fundamental Theorem of Calculus in one variable while treating others as parameters.
The condition that the domain is simply connected — containing no "holes" — is crucial. A vector field with zero curl on a domain with holes (like R² minus the origin) may fail to be conservative globally, even though it locally looks like a gradient field. The classic example is F = ⟨−y, x⟩/(x² + y²), which has zero curl away from the origin but whose line integral around a circle enclosing the origin is nonzero. This is why simply-connected domains are required in the equivalence theorem: holes allow closed paths that cannot be contracted to a point, which is exactly the topological obstruction to path-independence.