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.

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, Mars & 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 Mars-Rahu Conjunction - This is one conjunction which largely works based on the position and dignity of Mars. It is because the main function of Rahu is to take up the traits of other planet and blow them up. So, it is very necessary for Mars to have good dignity here and Rahu takes up the good traits of Mars. If Mars is well placed then it shows someone who is unbelievably courageous, up to the extent of Audacious. It also shows someone who is highly action oriented and can't sit still. It is like a whole power house is working inside the person. And the energy will go into the area of life which is represented by the house/sign where Mars-Rahu sits. But if Mars dignity is weak then it can lead to a situation where person can be anything from a habitual offender to serial killer. This would also show someone who is unable to handle stress, anger or aggression in life. This person may be argumentative but he will more get into physical fights and struggles. In both situations, person is very hardworking but in later case, person has no idea about the right direction in which actions should go. They can be good as Soldier, Fighters, Athletes but even for that Mars needs to be dignified. So overall, this conjunction is based on Mars's dignity. 

Who will win the Conjunction? - Mars will, as Rahu is just exploding what it is getting from Mars. So, Mars is dictating terms here. If Mars is good then Rahu is good else vice-versa.

Dignity & Strength - 

This conjunction is best in Capricorn where Mars is exalted and Rahu will blow up the results of an exalted Mars. Here, person will be very hard working and will always be working under Law. 

It can be worst in Cancer as Mars is debilitated and Rahu will explode the debilitated Mars results. Here, person can have anger issues and may easily get into physical fights. 

In Aries, Leo, Sagittarius and Pisces, Mars will be in very good dignity and Rahu will make Mars more strong. Such person will be courageous and active. He will be high on will power and actions will be well directed. 

In Scorpio, although Mars is in own sign but Rahu's presence means that person will have some crazy ups/downs in life. Only thing is that Mars presence will give courage and will-power to deal with all the ups/downs. This person can be extremely good with Occult Science. 

In Gemini, Virgo, and Aquarius, Mars is in enemy sign, hence person may not have right idea or direction of taking actions. He can have anger issues and Rahu is going to blow these traits out of proportions. 

In Taurus & Libra, Mars is in neutral dignity. Neither too good and nor too bad. So, Rahu-Mars conjunction in these signs will make person hardworking towards Ajeeva karakas related to house/sign but hurt the Jeeva Karakas of same house/sign, as it will do in any house/sign.