How to get a node by its url alias? I know I can load a node by its title with the node_load() function. Is it possible to load a node by its URL alias?
You want to investigate menu_get_object and drupal_lookup_path.

Something like this would do what you want:

$alias = "some/path/to/a/page";
$path = drupal_lookup_path("source", $alias);
$node = menu_get_object("node", 1, $path);