Step 1. Create a Custom Setting (Setup > Develop > Custom Settings) that you'd like to use. Create as many number fields as you need for the various numbers you'd like to track.

Step 2. Create a trigger (Setup > Customize/Create > [Object] > Triggers) that runs "before insert". The code might look like this:

trigger updateAutoNumber on myObject (before insert) {
  // Provide default values if settings do not exist.
  myCustomSetting__c settings = new myCustomSetting__c(name='myObject',Counter1__c=0,Counter2__c=0,Counter3__c=0);
  // Try and load the custom settings.
  for(myCustomSetting__c setting:[select id,name,Counter1__c,Counter2__c,Counter3__c from myCustomSetting__c where name='myObject'])
    settings = setting;
  // Loop through the records in the trigger.
  for(myObject record:Trigger.new) {
    if(record.CounterType__c == 'A') // whatever condition you'd like to use here.
      record.Name = 'A' + String.valueOf(setting.Counter1__c++);
    if(record.CounterType__c == 'B') // Another condition here.
      record.Name = 'B' + String.valueOf(setting.Counter2__c++);
    if(record.CounterType__c == 'C') // Yet another condition here.
      record.Name = 'C' + String.valueOf(setting.Counter3__c++);
  }
  upsert settings;
}

Note that you'll need to consider if you want to format with padding characters, etc, and of course changing the setting values will change the next auto-increment number (which could lead to duplicate names). Finally, you'll need to provide a page override to the New button for the object to default the name of the record to some value (typically we use something like "[automatically created when saving]"). This is typically done with a simple Visualforce page with the appropriate type of override:

<apex:page standardController="myObject">
<script>
window.top.location.href = '{!URLFOR($Action.myObject.New,null,[Name="[automatically created when saving]"],true)}'
</script>
</apex:page>

You will probably want to look at "Help & Training" for the correct usage of URLFOR. Finally, all you need to do is to create a test method to deploy the code to Production, and you'll be set. Please note that the object's Name field must be set to Text, not Auto-Number. In particular, some standard objects have auto-number values as the Name field type, and you won't be able to use this trick on those types of objects, since the field is read-only.