The primary goal of Ballroom Dancing (partnership dancing) is to become "ONE" on the dance floor by taking two, two-legged animals (dancers) and molding them into a single, "Two-headed Four-legged animal" (partnership). The dance connections are what allow the two-legged dancer to become a four-legged dancer.
The untrained dancer usually considers only one type of connection (the frame), but in reality there are many. Trying to lead the lady using only the frame creates uncomfortable, confusing and awkward partnership movements. The leader can control the movement of very specific parts of the lady through a number of various connections to create a comfortable, obvious and fluid partnership movement. I'm sure you have heard dancers or instructors say: "Connected to the floor", "Connected to the music", "Connected to your partner", etc.
"Connected to the floor" refers to the use of the floor by the supporting leg to initiate movement (not trying to move from free leg). The partner can only feel movement from the supporting leg. Untrained dancers try to move from the free leg to try not to step on the lady but this just makes the situation worse.
"Moving to the music" refers to coordinating the partnerships' physical movement to the music. Some examples are: Controlling the partnerships weight change over two beats for a "Slow" step, initiating a "Syncopated" movement, initiating a "Hovering" action, etc. All of this is based on the tempo of the music for this specific dance.
"Connected to your partner" refers to the physical connection between the two partners. The physical connection is the link that enables you to feel your partner. Examples of the physical connection are the frame, the hand in an "Under Arm Turn", the legs in a "Contra Head Check" element, etc. You want as much physical connection as possible, yet not so much as to be restrictive to the movement of the two-headed four-legged animal. The physical connection is always equal and opposite and it is pliable not frozen or stiff.
However, there are "visual connections" as well. The "visual connection" is what the partner actually sees. The head turn element in a "Link to Promenade" is an example. Part of the lead is the physical rotation of the leader toward the lady and the second part of the lead is the lady visually seeing the leaders head turn to the left. Solo turns such as in the "Open Side Lock" pattern and the "Walk Around Turn" have visual lead elements.
"Invisible connections" exist too. The "invisible connection" is a logical connection that is developed through a physical connection. The connection from the frame to the center of the partnership to initiate movement is an "invisible" connection. The ladies crossing leg movement in a "5th Position Break" element and the ladies leg movement in many "Ronda's" are controlled by the "invisible connection".
The connections, other than the frame, are the major parts missing from the "Pattern Dancer". A "Pattern Dancer" is a dancer (leader or follower) who has learned his or her dancing from "Group Classes". Group classes teach primarily "Foot Patterns". A mentality is developed in "Group Classes" that each partner dances his or her part of the pattern. If the lady doesn't know that particular pattern, she will have a hard time dancing it gracefully. The more different dancers she dances with, the more likely this will occur. Adding connections to the partnership will solve this problem by creating a lead and follow environment.
How does a dancer learn how to create and use this myriad of "Connections"? The answer is to become a "Trained Dancer". The" Trained Dancer" is one that develops an understanding of how the "two-headed four-legged animal" works. I don't know of any other way except for "private lessons". There is so much of being a "Trained Dancer" that is based on feeling. The only way to get feeling is through physical contact with your instructor in a private lesson. It is often very hard to verbalize the feeling, but once your instructor demonstrates the feeling, you will know what is trying to be communicated. The "Trained Dancer" (leader) knows the ladies part as well as his own. He knows what kind of action he is trying to get the lady to execute. The "Trained Dancer" (lady) knows how to recognize, maintain and respond to the connections". The connections are the developed by both the leader and lady. The "Trained Dancer" grows over time. It just doesn't happen over night. It is a process.
"Connections" are the primary elements to four-legged movement. The "connections" are what separates the two-legged from the four-legged, the "Pattern Dancer" from the partnership dancer, the difference between the "Trained Dancer" and the "Untrained Dancers". The goal of being "One" on the dance floor is achievable. Allow your "Trained Dancer" to evolve.