Morse theory is another good example. Indeed it is the inspiration for Floer theory, which has already been mentioned.
Atiyah-Bott's paper "Yang-Mills equations on a Riemann surface" and Hitchin's paper "Self-duality equations on a Riemann surface" both contain rather striking applications of Morse theory. The former paper contains for example many computations about cohomology rings of moduli spaces of holomorphic vector bundles over Riemann surfaces; the latter paper proves for instance that moduli spaces of Higgs bundles over Riemann surfaces are hyperkähler.
Note that these moduli spaces are algebraic varieties and can be (and are) studied purely from the viewpoint of algebraic geometry. But if we look at things from an analytic point of view, and we realize these moduli spaces as quotients of infinite dimensional spaces by infinite dimensional groups, and we use the tools of analysis and Morse theory, as well as ideas from physics(!!!), then we can discover perhaps more about these spaces than if we viewed them just algebraically, as simply being algebraic varieties.