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.

Saturn - It is all that which we don't like in our life. Saturn is Delay, Frustration, Restriction, Limitation and Anxiety. It is people of Authority, like Boss at your Work Place or Father-Figure at Home. Saturn is our Karmic Debt from past-life. Saturn is the things we didn't treat too well in our past life. Saturn is our Challenges and Lessons of life. Saturn is Old-Age, Teeth/Bones in Human Body, Diseases, Changes etc. Saturn represents our Step-Relations, it means that wherever Saturn sits, people related with that house give us Step Treatment. Saturn is Hard Work, Efforts and most importantly Perseverance etc. Nothing comes easy and in one go with Saturn. You have to work hard continuously over the longest possible period of life. Saturn makes a person Legend as Legends have one thing in common that they choose one field and work hard in it for whole life. 

House Ruler - Besides these karaka things, Moon & Saturn 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-Saturn Conjunction - This is one of the most stressful conjunction one can have. Moon is Mind, Mother and Emotions. Saturn is Stress. So, simply speaking stress is coming to mind. This person can have very stressful 1st half of life. It shows some deep emotional issues. It shows someone who had to repress his emotions in early life. It also shows that person either had to stay away from Mother or Mother was just too strict. Either ways, it shows a lack of nourishment from Mother. As they are not given Mother's care from beginning, it develops them as people who are unable to express their emotions correctly. At the same time, it shows a mind which feels burdened by responsibility and duty from early life. In early life, this conjunction can even lead person into gloom and depression. This person is seen as overall a very serious person by people. They can be very dutiful and law abiding. They grow-up in life at a very early age due to the responsibilities they take from early age. But in the end, same seriousness from early age helps them in becoming a legend as no legend of any field ever had a happy early life. One place where this conjunction can be very useful is Meditation. Moon represents Mind & Thoughts and Saturn is Limitation. So, this conjunction shows someone who can limit his thoughts easily. So, it shows someone who can meditate easily.

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

Dignity & Strength - 

In any sign or any house,it will be a tough conjunction. It can be worse if Saturn is debilitated (in Aries) or in Enemy Signs (Cancer, Scorpio & Leo) but it can never be Rosy. In Cancer & Scorpio, it shows some huge issues with Mother in life.