Static pods are pods created and managed by kubelet daemon on a specific node without API server observing them. If the static pod crashes, kubelet restarts them. Control plane is not involved in lifecycle of static pod. Kubelet also tries to create a mirror pod on the kubernetes api server for each static pod so that the static pods are visible i.e., when you do kubectl get pod for example, the mirror object of static pod is also listed.
You almost never have to deal with static pods. Static pods are usually used by software bootstrapping kubernetes itself. For example, kubeadm uses static pods to bringup kubernetes control plane components like api-server, controller-manager as static pods. kubelet can watch a directory on the host file system (configured using --pod-manifest-path argument to kubelet) or sync pod manifests from a web url periodically (configured using --manifest-url argument to kubelet). When kubeadm is bringing up kubernetes control plane, it generates pod manifests for api-server, controller-manager in a directory which kubelet is monitoring. Then kubelet brings up these control plane components as static pods.