I must admit that while partner connection is already extremely difficult to explain, demonstrate and acquire in actual dance classes, it is a hundred  times more difficult to simply explain in words. But we will still try, together with the help of some dance friends. There is much information about this topic in Mark Balzer's FAQ list on Lead and Follow. So please make sure to link up for a better understanding.

     Partner connection involves and requires among others good hand and arm tone, dance frame, and posture, as well as a working understanding of the dynamics of body movement, weight changes, individual balancing and common centering, force level leading and following techniques.
     In the Philippines, those teachers and dance instructors who have enough patience and knowledge to teach lead/follow to their students and guests have referred to these as simply leads, and more specifically as chest leads, arm leads, and hand leads. As far as I understand, these are all conditions, signals and signs which a dancer uses to communicate with the partner. Lest I just repeat what other have already said, please consider the following links to other sites.
 

POSTURE
     Generally, dances require an erect yet relaxed body posture from both the leader and the follower. The key word is relaxed. However, several dancers change the overall look and line of the couple with a slight forward bend, such as in East Coast Swing (Boogie). More insights from FAQ on Posture.

WEIGHT CHANGES
     A weight change simple means the positioning of the dancer's Center Point of Balance (CPB) over one of either foot. It is definitely not a point, or tap. The body weight is transferred firmly on the center of the foot, precisely on that area below the toe and beside the ball of the foot. Once the CPB is precisely on top of this point, the dancer will know he has actually stepped on that foot and is totally balanced. Check out FAQ on Weight Changes.

DANCE FRAME
     A well-toned dance frame, balanced and stable, firm and solid, is a result of good posture and is the  key element which enables the dancing couple to move around a common center. It also helps the partners to be concentrate on the lead/follow signals and messages instead of the other unnecessary body movements (noise). A more detailed discussion is available from the FAQ list on Dance Frame.

BODY LEADS
    We're said it several times before. All leads emanate from the body, thus you hear of teachers referring to body leads. Even if the only physical connection between you and your partner is through the hands, the source of the lead is still the  body. To be more precise, the source is the solar plexus, or the diaphragm, that small round muscle at the bottom of the ribs. In dancing, this is known as the Center Point of Balance or CPB. The chest, arms, hand and fingers, are mere extensions of the body, through which the lead is coursed. Because of this, the partners should always maintain a clear and sensitive connection with each other, so that each lead can easily be read by the follower, and the follower can react on the intended time with the intended step and position in the intended direction. See FAQ on Body Leads.

ARM TONE
   Arm tone, is the amount of force or pressure used by the dancer's arm to send and receive messages and signals to and from a partner while dancing. Firm, but not forceful, tension is most appropriate. Various dance movements and positions will require a variety of pressure levels; and so does dancing with various partners. Some partners are not yet sensitive enough to subtle leads, and require stronger arm tension to they can feel the connection. Over time however, they will learn to tone these down as their sensitivity heightens. See also FAQs on Arm Tone and Shoulder and Arm Leads.

HAND TONE
   Hand tone is the level of force or pressure of the hand and fingers which a dancer (leader or follower) uses to send and receive messages to and from a partner while dancing. The rule my partner and I use is simple: firm yet flexible. The hold should not be a grip but a relaxed touch. So why the lady's hand may be in the man's hand, neither hand grips the other. With the lady holding her own arm's weight, her hand would be comfortable light in his hand. The same hold applies to her left hand on his shoulder or upper arm, and his right hand on her back. The hold allows from the flexibility required for turns and changes in dance positions of the partners. See also FAQs on Hand Tone and Hand and Finger Leads.

FORCE LEVEL
     This simply means that the leader should not use too much force -- like pushing, pulling and hauling -- nor too little force as to be indistinguishable from a touch when leading the partner. Dancing is meant to be pleasurable, not the cause of sprains and bruises, and worse quarrels and fights. A better discussion is available from FAQ on Force Level and FAQ on why Forceful Leads are not Required.

A much more extensive discussion on partner connection may be found in Dance FAQs.