A goal counts when the puck fully enters the goal trough and touches the back or the bottom of that volume. Equipment.
The following shots would be considered a goal.
The following shots would not be considered a goal.
No-goal
Sometimes the puck goes in and still does not count. If attackers fouled on the way in (or as it went in), the ref can signal no-goal, wipe the score, and set an advantage restart the other way.
If defenders foul and you still score, the goal usually stands: they gained nothing by fouling. See Fouls and Atlantis — No-goal calls.
Around the goal
Defenders will put as many bodies down as they can. Match that with offensive players: position players at the ends of the goal to “seal” the goal. Players positioned in the middle of the goal can grab pucks pushed to the middle by the folks “sealing the goal” so clears die in the middle.
Keep possession even when the first shove does not go in; a lost puck in front of their goal is how breakaway goals start the other way.