Sending alerts to users are much easier and administrator can assign active directory user for dispatching emails.
Please note that share point uses email address from user’s profile for dispatching emails. Therefore alert cannot be configured for user if email address is not associated in active directory account.
Coming back to original points sending alerts to groups can be of two types.
1. Sending alert to AD Group
2. Sending alert to Share point Group
For sending email to AD Group you need to make sure two things.
1. Group needs to be “email enabled security group”. As distribution lists are not supported in share point
2. You need to explicitly add group in permission setting in list/Document Library etc
There is one limitation is that you cannot edit group and can only delete/recreate alert from Site Action->Site Setting ->User Alert (Site Administration) option
For Sending email to Share point Group
There is no out of box support for sending email alerts to share point group.
However there is a free solution for sending emails to group at codeplex that can be utilized for this purpose.
http://advancedalert.codeplex.com/ (This will not update alert when new user added in groups)
Few solutions are also available for purchase e.g.http://www.sharepointboost.com/alertbuynow.html