What do I do after the bounding box test against a tile to determine whether there is a real collision against the contents of that tile? And if there is, how should I move the object in response to that collision?

I have a small object, and test for collisions against the tiles that each corner of it is on.

Here's my current code, which I run for each of those (up to) four tiles:

// get the bounding box of the object, in world space
objectBounds = object->bounds + object->position;  
if ( (objectBounds.right >= tileBounds.left) && 
     (objectBounds.left <= tileBounds.right) && 
     (objectBounds.top >= tileBounds.bottom) && 
     (objectBounds.bottom <= tileBounds.top))
{
    // perform specific test to see if it's a left, top , bottom 
    // or right collision. If so, I check to see the nature of it 
    // and where I need to place the object to respond to that collision... 
    // [THIS IS THE PART THAT NEEDS WORK]
    //
    if( lastkey==keydown[right] && ((objectBounds.right >= tileBounds.left) && 
                                   (objectBounds.right <= tileBounds.right) && 
                                   (objectBounds.bottom >= tileBounds.bottom) && 
                                   (objectBounds.bottom <= tileBounds.top)) )
    {
        object->position.x = tileBounds.left - objectBounds.width;
    }
    // etc.