OK, I've cleaned up the code a bit, I will explain below.  This is just the for loop and right after:

for (Task t: Trigger.new){
	if (t.accountID != NULL){
	  acctIDs.add(t.accountID);
//Create a map to match the task related to ID's with their corresponding account ID's
	  Map<ID, Account> acctMap = new Map<ID, Account> ([Select ID, Last_SW_Activity__c from Account where ID in :acctIDs]);
//Create the account object
      Account acctRec = acctMap.get(t.accountID);

//If the account ID isn't null, the subject line starts with "sw", and the task has been marked as completed
//Check to see if the Last_SW_Activity__c field is current compared with the latest completed activity    
	If (((t.subject.indexOf('sw')==0) && (t.Status == 'Completed')) && (acctMap.get(t.accountID).Last_SW_Activity__c < t.LastModifiedDate || acctMap.get(t.accountID).Last_SW_Activity__c ==null)){
//Update the Last_SW_Activity__c field on the account object with the task's end date  
  		  acctrec.Last_SW_Activity__c = t.LastModifiedDate;
  	}
      }
  }
  update acctRec;

 I have made three main changes:

1.  You had 2 if statements one after the other and I combined them into one.  You can leave it the way you had it and it should work, but this looks a little cleaner and can be easier to read.

2.  I moved this check: if(t.accountID != NULL), to the top of the for loop.  Essentially, if there is no associated account we are just ignoring the rest of the code in that iteration of the loop.

3.  I moved update acctRec outside of the for loop.  When you call Update (or insert, or delete for that matter) on a collection, in this case a map, it will update all the records in that collection.  So you were updating every account each iteration.  This will let you run into governor limits with bulk processing.  So now, you will be just changing all the records in your for loop and then updating them all once, at the same time.

As for the test, here is the logic it should follow:

1. Create and insert an account

2. Create and insert a task associating it with the account.

3. Query the Account from the database, this should give you the updated value in Last_SW_Activity__c

4. Do the same for the task

5. Use system.assertEquals or similar to check and see if the task that you queryed has the same value in lastmodifieddate as the account does in Last_SW_Activity__c.

That will give you the test coverage that you need, but to be through you should check two more cases with similar logic:

1.  Insert 200 tasks instead of one - will check for bulk inserts/updates

2.  Insert a task that is not associated with an account and make sure an exception is not thrown.

Although these tests are not necessary to get the code working, they may serve you well later by ensuring that your code can not be broken by a later update.