For built in signals the signal is automatically emitted when the event occurs and the function that you assigned to be the callback via connect()
(or in editor) will be called. If you want to emit your own custom signal when a mouse enters your Area2D
simply emit it in the callback:
extends Area2D
signal my_custom_signal
func _ready() -> void:
self.connect("mouse_entered", self, "_on_Area2D_mouse_entered")
func _on_Area2D_mouse_entered() -> void:
print("I'm automatically called when a mouse enters my Area2D.")
emit_signal("my_custom_signal")