What is responsive mode in the context of research funding?
I'm answering within the context of UK Research Council funding:

Some Research Council funding is allocated on the basis of specific calls for proposals (aka managed mode): so for example there may be a call to set up a Centre in End-Use Energy Demand, and universities (or consortia of universities) will shape proposals to deliver a Centre in End-Use Energy Demand. As part of the call, criteria will be defined about the sort of things such a Centre should deliver, that will be more specific than the usual - impact, engagement, publications - though they will not go as far as specifying particular research questions. Bid applications will be assessed against the usual generic quality and value-for-money criteria, but also against how well they meet the specific proposal criteria.

However, other Research Council funding is allocated in responsive mode: these proposals are driven wholly by what the applying Principal Investigator wants to research, and as long as it is within the remit of the Research Council, it will be considered, and assessed on its own merits. That is, the assessment will be against general criteria, and whether it is at all on-topic or not for that Research Council; but assessment will not be against how well it meets a pre-defined specification for a narrow research area, because for responsive-mode, that pre-defined specification does not exist.

Here's some guidance from the Biotechnology and Biological Sciences Research Council (BBSRC): 


  Grants are awarded in response to unsolicited research proposals in any area relevant to BBSRC’s mission. As a general principle, BBSRC allocates as much research grant funding as possible in responsive mode. You can apply for a variety of investigator-led research activities through responsive mode.  ...
  Applications are assessed by the appropriate BBSRC Research Committee using the standard BBSRC competitiveness appraisal factors.


Responsive mode funding is only available to Higher Education Institutes and Research Institutes, whereas the funding in calls for proposals (managed mode) is available not only to those, but in addition it is also available to other Independent Research Organisations