For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex nonsingular plane cubics $X_t$ degenerating to a nodal cubic $X_0$. It is possible to understand the change in topology rather explicitly.
You can set up a basis of (real) curves $\alpha_t,\beta_t\in H_1(X_t,\mathbb{Z})$, so that $\beta_t\to \beta_0\in H_1(X_0)$ and $\alpha_t\to 0$ as $t\to 0$. So that $\alpha_t$ literally is a vanishing cycle. There is more. As you transport these cycles around a loop in the $t$-plane, you end up with a new basis $T(\alpha_t),T(\beta_t)$. This is related to the old basis by the Picard-Lefschetz formula $$T(\alpha_t)=\alpha_t$$ $$T(\beta_t) = \beta_t \pm (\alpha_t\cdot\beta_t)\alpha_t$$ You visualize this by cutting along $\alpha_t$ and giving a twist before regluing.
You can jazz this up in various ways of course, and this is the modern theory of vanishing cycles. In modern notation you'll see a nearby cycle functor $R\Psi$, which corresponds roughly to $H^*(X_t) = H_*(X_t)^*$ and a vanishing cycle functor $R\Phi$ which measures the difference between $H^*(X_t)$ and $H^*(X_0)$.