If on term-page -> get the current term?

You’ll want get_queried_object(). This is a very generic function – and simply returns the queried object- so a single post, this would be a post object.

For instance, the return object may be of the form:

Object (
    [term_id] => 299
    [name] => test
    [slug] => test
    [term_group] => 0
    [term_taxonomy_id] => 317
    [taxonomy] => event-category
    If on term-page -> get the current term? => 
    [parent] => 0
    [count] => 2
)

So for instance:

  $q_object = get_queried_object();
  if( isset($q_object->taxonomy) ){
     $taxonomy = $q_object->taxonomy;
  }else{
    //Not a taxonomy page
  }

To use this in a function:

function wpse51753_breadcrumbs(){
    if( !is_tax() && !is_tag() && !is_category() )
       return;

    //If we got this far we are on a taxonomy-term page
    // (or a tag-term or category-term page)
    $taxonomy = get_query_var( 'taxonomy' );
    $queried_object = get_queried_object();
    $term_id =  (int) $queried_object->term_id;

    //Echo breadcrumbs
}

Then just wpse51753_breadcrumbs(); in your template wherever you want to display the breadcrumbs.