There are 5 widely-invoked modes of convergence for a random variable, listed below from strongest to weakest:
1. Complete convergence
$\sum_{n=1}^{\infty} P \left( | X_n - X | > \epsilon \right) < \infty$
2. Almost sure convergence
$P \left( \lim_{n \to \infty} X_n = X \right) = 1$
3. Convergence in $r^{th}$ mean
$\lim_{n \to \infty} E \big[ \;| X_n - X |^r \; \big] = 0$
4. Convergence in probability
$\lim_{n \to \infty} P \left( | X_n - c | > \epsilon \right) = 0$
5. Convergence in distribution
$\lim_{n \to \infty} F_{X_n} = F_{X}$ at all continuity points
I'm seeking (to whatever extent possible) intuitive motivations for, and connections between, these modes of convergence. I know how to prove their interrelationships formally, but I find these concepts rather abstract and would like to develop a more well-rounded understanding. I often find visual and/or simulation-based intuitions especially helpful, but use your creative pedagogy.