CURLMcode curl_multi_perform(CURLM *multi_handle, int *running_handles);
When you call curl_multi_perform() and the amount of running_handles is changed from the previous call (or is less than the amount of easy handles you've added to the multi handle), you know that there is one or more transfers less "running". You can then call curl_multi_info_read(3) to get information about each individual completed transfer, and that returned info includes CURLcode and more.
When running_handles is set to zero (0) on the return of this function, there is no longer any transfers in progress.
If you receive CURLM_CALL_MULTI_PERFORM, this basically means that you should call curl_multi_perform again, before you select() on more actions. You don't have to do it immediately, but the return code means that libcurl may have more data available to return or that there may be more data to send off before it is "satisfied". Do note that curl_multi_perform(3) will return CURLM_CALL_MULTI_PERFORM only when it wants to be called again immediately. When things are fine and there is nothing immediate it wants done, it'll return CURLM_OK and you need to wait for "action" and then call this function again.
NOTE that this only returns errors etc regarding the whole multi stack. Problems still might have occurred on individual transfers even when this function returns CURLM_OK.