Let's 1st understand what these planets represent on their own?

Meaning of Moon-Mars Conjunction - First of all, this conjunction is known as Chadra-Mangal Yoga. It is a good yoga to have if both are in good dignity as Moon and Mars are natural friends but this conjunction can be quite worrisome at most of the times. Reason is simple that Moon remains the most gentle planet ever and now it is conjunct with a natural malefic like Mars. So, Moon/Mind gets severely impacted by the malefic trademark nature of aggression and anger of Mars. So, this person can be easily angered or he can take actions in the heated moment of anger without applying much mind, if Mars is in wrong dignity. Agreed that Mars can make a person very action oriented or energetic but to act in right direction, you need Mars in good position else this conjunction can lead you into violence too. This also shows someone's mother can be very aggressive or active. If Mars is in good dignity, then this conjunction can work well as mind will be directed towards right & dignified action. In that case, person will be highly goal oriented and will reach great heights in life based on own actions. Else closer the degrees of Moon & Mars, the more agitated a person can feel. In such situation, meditation can be of great help or if person can work with himself in setting goals for himself, he can get good results of this conjunction. Key here is to divert the anger in positive means. If a girl has this conjunction then she can be tomboyish. 

Who will win the Conjunction? - Of course, Mars will. Moon is too gentle to impact Mars in anyways. So, Mars will have upper hand in this conjunction at any time. 

Dignity & Strength - 

This conjunction will be best in Aries, Leo, Sagittarius and Pisces as both planets will be well placed in good dignity. So, person will have all the idea and inspiration to act in a dignified way. 

Mars will over power Moon easily in Scorpio & Capricorn as Moon is weak in these signs. 

In Cancer, it can be really dangerous as although Moon is in own sign but Mars is debilitated. Although a Neech Bhang Raaj Yoga is created but it helps in professional life not in personal life. So, at personal level, he can have some real anger issues. 

In Taurus & Libra, it is a neutral conjunction and depends on other factors of chart for good/bad results. 

In Gemini & Virgo, it is not a good conjunction to have as both Mars & Moon are weak in dignity. Another position where person can have serious temper issues.