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

Mars - Mars represents our will power, courage, ability to take actions, aggressive nature, anger, our fighting ability, brother, male friends, a boy friend for a girl, a soldier, an athlete and real estate 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, Mars & 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 Mars-Saturn Conjunction - This is one of the most stressful conjunction one can get as Mars & Saturn are 2 strong malefic and they are mutual enemies too. So, there is a planetary war going on in the house where this conjunction falls. That house is like a War Zone. Mars has a nature to be initiative and aggressively go towards his actions, complete it and then move towards next task but Saturn has a nature of slowing down things & make someone put best efforts over a long time. So, Mars is all about finishing the job quickly but Saturn has nothing to do with quickness. This creates a stress. So, in whichever house this conjunction falls, person will be 1st looking eagerly to get results of things related to that house, but Saturn limits that. Then person has to put repetitive efforts to get results. Hence, Delays and Frustration. But end result is that person becomes very hard working. This is the biggest benefit of this conjunction. As nothing comes easily or early in life, these people become very hard working in life and specially related to things where this conjunction falls. Also, Mars is Soldier and Saturn is Law. So, many a times they are seen in Law field. If their dignity is good then person can be very law abiding or else a rebel. At the same time, both these planets represent Technology or Machinery, so they can be very technology oriented people. If their dignity is not good then this conjunction can also lead to accidents as there is conflict between speed of Mars and slowness of Saturn. So, they can be accident prone. Overall, the keywords for this conjunction is Hard Work and Perseverance. They will get only after delay and best efforts over a long period of time. 

Who will win the Conjunction? - Saturn will, as it will stop the result of Mars for a long time and make Mars work hard to achieve things. 

Dignity & Strength - 

This conjunction is best in Capricorn where Mars is exalted and Saturn is in own sign. 

It can be worst in Cancer as Mars is debilitated and Saturn is in enemy sign. In such cases, support of other planets like Jupiter is much needed to make this conjunction work. 

In Aries, Leo, Scorpio, Sagittarius and Pisces, Mars will be in better dignity than Saturn so actions of Mars will be dignified but Saturn's impact will be as is, i.e. of delay and frustration. Saturn in Aries person can have weak bones. 

In Taurus, Gemini, Virgo, Libra and Aquarius, Saturn will be in better dignity. Anyways, Saturn is driving the conjunction. These positions will give further strength to Saturn.