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

Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.

Rahu - It represents Illusion, Foreign Lands, Foreign Things (foreign as in unknown things. So, for someone living in Punjab, culture and rituals of Kerala is also unknown and hence Foreign), Drugs, Medicines, any type of intoxication, Fame, Wealth, Success, Obsession, Past-life incomplete Desires, Movies, Television, Online World, Cheating, Imagined Fears, Unusual Things, Unique Things, Creativity, Rule breaker. Rahu also blows things out of proportions

House Ruler - Besides these karaka things, Moon & Rahu 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 Moon-Rahu Conjunction - Moon is Mind & Emotions. Rahu is Obsession. So, this conjunctions simply means someone who is obsessed for emotions or highly/obsessively emotional. In common medical term, this conjunction shows someone who has OCD - Obsessive Compulsive Disorder. When they want something, they will never listen to any counselling. They will try their idiotic best to get that thing. It doesn't matter to them at that moment as to what & how much it is going to cost them? This conjunction is similar to someone who has Rahu in Cancer, i.e. obsessively emotional. If they are happy, they can throw party to whole town. If they are sad then they can even contemplate suicide. It gives a very unusual or unique relation with Mother. They can have some illusions regarding their Mother. It can also be the case that they are taken care by Foster Parents, especially Mother. They themselves can have Motherhood through unusual ways like Test Tube Baby, Adoption or Surrogacy etc. Their highly emotional nature can lead to Caring Nature too where they can be very caring/emotional towards others. If they go in field of caring/healing others, it can help their Moon. But Rahu is still illusion and it is with its enemy Moon. So, this person can be equally illusive, i.e. Cheat or Deceitful. It can give an inclination to find peace in foreign lands or by some very unique/unusual creative ways. It is a difficult conjunction. The only way out here is to Meditate. 

Who will win the Conjunction? - Rahu will, in any case.

Dignity & Strength - 

In any sign or any house,it will be a tough conjunction for Moon. In Taurus only, it can be little better but person will be obsessed towards Wealth and Assets. So, it is a different type of madness but only thing in favor is Moon exalted. So,Mind will have some balance. In other signs, it is all Rahu's impact.