Social dancers and teachers emphasize the importance of connection in partnering. There are two kinds of connection—physical and mental. Some teachers stop after working on just the physical connection but I think both are essential, even inseparable, in the holistic mind-body-spirit connection of dance partnering.
The physical connection
Connection is how dancing feels to you and your partner, not how it looks.
The technique of connection in the frame of closed dance position ("waltz position") varies widely depending on the dance form, from waltz to tango to swing. The physical connection of blues versus competition ballroom dance are at the opposite ends of a broad spectrum. But in general, when you and your partner are in closed dance position, hold with comfortably braced hands, without clamping, and somewhat firm arms, without being rigid. In some partnered dances (but not all) the Lead's left and Follow's right hands are lightly braced into each other. When the Lead compresses a little more, the Follow responds by bracing back to match, showing an "I'm ready for anything!" confidence. The physical connection of social dancing is balanced and reciprocal.
Several dance forms don't have braced hands. Instead, the Follow is cradled into the Lead's right arm. Either way, braced hands or not, lead from the core of the Lead's body to the Follow's core, with a body-lead, instead of poking, pulling or twisting with pressure points. Follows, respond from your core, balanced and reciprocal.
Swing dancing is often connected with one or two hands. Your hands are comfortably compressing/pushing or catching/pulling your partner's hands, without squeezing like a clamp, while your arms are fairly firm, without either rigidity or noodle arms. Think water skiing when pulling, because your pull is connected to the core of your body (as is your push). Try to exactly match your partner's compression and tension, balanced and reciprocal.
The three briefest descriptions of the mental connection are:
● Your partner is a living, caring person with feelings, not a practice object to assist your learning. Treat your partner with kindness, consideration and your full attention. In a dance class, try to help your partners improve as much as you would like them to help you.
● The best mental connection to your partner is empathy. Empathy can be as simple as having an awareness of how your dancing must feel to your partner, from their perspective. The goal of this empathetic connection is to improve your partner's experience, perhaps with clearer and more comfortable leading, more compassionate following, or with a reassuring smile.
● Connection is ideally a two-way conversation. Leads remain aware of the Follow's interpretation choices, firmness or lightness of the physical connection, styling, and innovations. Leads and Follows constantly adapt and adjust to reciprocate in this balance. Everyone dances; no one gets danced.
Musicality is another part of the mental connection, in the three-way dynamic between the Lead, the Follow, and the music. But this is a long discussion in itself and I'm keeping this page as brief as possible.
Again, connection is how dancing feels to you and your partner—physically, emotionally and musically—not how it looks. When you help your partner have fun, you're also having more fun.
More suggestions for improving your mental connection are here (perhaps the page you just came from).