So finally, first of the Conjunction articles.
Let's 1st understand what these planets represent on their own?
Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.
Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.
House Ruler - Besides these karaka things, Sun & Moon will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include those things at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.
Meaning of Sun-Moon Conjunction - First of all, if Sun-Moon are conjunct in a chart then it means that person is either born on New Moon day or very near to it. New Moon can also be seen as No Moon day or Amavasya. As Moon represents Mother, it shows the absence of Mother or Motherly Nourishment in Life for various reasons. Even if Mother is present, she can be very strict and kind of disciplinarian. Person himself can have lack of emotions and will be all about rules & regulations. If Moon is too close to Sun degree-wise then it is combust and these results can be easily evident. Sun is Soul and Moon is Mind, so good thing is that person's soul & mind are aligned towards a common goal. There is no dilemma for person as to what he wants to do in life. It may not be the rightest thing in world to do but person has no confusion. As both planets represent Govt, it is a good position for benefits from Govt. In most of the cases, Emotional Side or Motherly Nourishment of person will suffer due to fire of Sun. Even if Mother is caring and emotional towards child, this conjunction can show a father who is totally dominant and Mother may suffer in her life. As Sun represents Career and Moon represents Nourishing & Taking Care of Others, they can go into Careers like Health Care & Teaching, but for that Moon should not be combust, else they will realize such path with delay in life by which time it may be too late to change the career.
Who will win the Conjunction? - Of course, Sun will. Moon is too gentle to impact Sun in anyways. Actually, Moon's existence came into being due to the light of Sun which fell upon Moon. So, it is almost impossible for Moon to over power Sun's fire.
Taurus & Cancer - Still, in signs like Taurus (exalted sign) & Cancer (own sign), Moon will be able to have some say in conjunction and in these signs, person may not be emotionless or he may get good support and upbringing from Mother. Remember, everything is in addition.
Aries and Leo - In signs like Aries (exalted sign) & Leo (own sign), Sun will have complete control over Moon as Sun is too strong here.
In signs like Capricorn, Aquarius Scorpio, & Libra, where they both are weak, Sun will command power over Moon due to its natural fire power. Moon will be impacted by Sun's presence in same sign/house, no matter what?
Even if Moon MD is running and they both are conjunct, we will see results of both Sun and Moon as it is impossible to neglect Sun if it is sitting in same house. Just imagine, Sun sitting besides you. You will always feel the heat. So, Sun-Moon conjunction is all about Sun's impact over Moon, rather than vice-versa.
Hope this helps. Please post any comment or query you may have.