For me, it feels like mutual interest, engagement in the interaction, and an ability to listen and understand what the other person is saying. It also has elements of palpable mutual respect. For me, a connection with someone does not equal to attachment at all, it's more about interest and involvement in the interaction. Often it is triggered by perceived similarities between myself and the other person as well, a sense that we have something in common and compatible in personalities.

Lack of connection is usually when I don't feel and targeted interest from the other, or I feel that they do not listen and just keep pushing their agendas. Also if we think and communicate in dramatically different ways, our interests don't have much overlap, etc.