Thanks for suggestion Vitaly.
It is possible to subscribe to each forum (and also individual topics). Click the subscribe forum link at the bottom of the forum you want for notification of new posts.
By default, forum will only send one notification email to each user subscribed to the topic when someone replies. Additional replies will not send more notifications UNTIL the user visits the topic again.
1) User subscribes to a topic
2) Someone replies to the topic and a notification email is sent out
4) Someone else replies to the topic, no notification email is sent
5) User visits the topic again
6) Someone replies to the topic, so another single email is sent out.
You manage your subscriptions from the User Control Panel.