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.

Mercury - Mercury represents communication skills, marketing, calculative ability, younger siblings, hobbies, skills with hands, intelligence, quick decision making, logical thinking, youthfulness etc.

House Ruler - Besides these karaka things, Moon & Mercury 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-Mercury Conjunction - These two are natural enemies to each other. So, this is rather not a very good conjunction to have. Only good thing attached here is that both are not malefic planets (Moon is benefic and Mercury is Neutral). So, even if they are together, there is no serious damage to house like if Saturn-Mars were together. Now, where it can create trouble? Moon is Mind and Mercury is Travelling, lol. So, this person's mind can never be still at one thing. It is always full of ideas. Before they start working on one idea, 10 other ideas can come to them. And that's where the problem is. They may not be able to take one decision and stick with it. It can be a good business mind with lots of ideas or good mind with lots of writing/creative ideas but person may not be able to stick with one decision for a long term to see the benefits of that decision. Of course, person will improve with age and maturity, but in early life it can be a real worry. Also, they can be very talkative and if both planets are in wrong dignity, majority of their talks can be worthless. At the same time, they can be very good at business, writing and communications. They can be very curious, research oriented and may like to collect lots of information. Mind will love to travel. As always, Moon is easily impacted by the nature of Mercury. If Mercury is well placed, then these people can be highly analytical, critical and perfectionist. They can be real clever or even cunning people. Meditation is best remedy for them but Meditating can be a real hard task for their wandering mind. So, taking control of ever running thoughts can be the biggest challenge for these people. They can do well in any field related with Communication, Counselling and Teaching. This conjunction also shows someone whose Mother can be a Counselor/Teacher or in the field of writing or travelling. 

Who will win the Conjunction? - Mercury will. Mind will be totally grasped by Mercury through its traits and nature. 

Dignity & Strength - 

This conjunction is best when Moon-Mercury are in Taurus as Moon is exalted and Mercury is in its best friend's sign. Here, Mind/Moon will find its balance and will be able to properly utilize the ideas of Mercury. 

Leo is another sign where they can work well together as both are in friendly sign. 

In most of the other signs, either of or both of planets are not in good dignity. 

In Aries, Cancer, Sagittarius and Pisces, Moon is well placed but Mercury is in enemy or debilitated sign. As Mercury is still impacting Moon, it shows person's mind is unable to manage wayward thoughts/ideas of Mercury. 

In Gemini, Virgo, Libra, Capricorn & Aquarius, Mercury is strong and Moon is in enemy signs. It shows that person's mind is becoming too analytical, critical and perfectionist. 

In Scorpio, both planets are in bad dignity. The only field where this conjunction can work is Occult and Mysticism as person can be very curious to know about secret stuff.