>>What's an ACOM message?<<
In this case it means a private message sent from one Ancestry.com (ACOM) member/user to another. At the upper right of your screen, there is an envelope icon, this will display a number, ex. (1), next to it, if you've been sent a private message.
>>All registered members to the site are able to send and receive messages through ACOM's message system.<<
Only ACOM paying subscribers can initiate a private message to another member, but any status member who receives a private message can respond to it.