The Markov property dictates that the future states of a stochastic process only depend on its current state, not any previous states. In a discrete setting, this can be written as:
In finance, we want asset prices like stocks to satisfy the Markov property: say the stock price was 100 last week and it's gone up by 10% to 110: the probability that it goes up by another 10% or that it goes down by 10% should not depend on what happened last week; it should only depend on the current state of the world.
The Markov property is also called "memorylessness": somewhat intuitively, asset's behaviour should not depend on the "path" it took to reach the current state; assets should not "remember" their past in order to drive their future states (if we could determine stock price based on the past, it would be easy to make money, wouldn't it? The fact that everyone is trying and most people fail over the long term suggests that indeed, real-world assets are Markovian).
The martingale property is important for derivative pricing, and is related to the cost of borrowing and lending money. Simply put, in "expectation" (where "expectation" represents the mathematical operator that sets the future arbitrage-free price), we want asset prices to equal to their current value compounded at the rate of borrowing money; indeed, as proven in the Fundamental Theorem of Asset Pricing.
PS: note that not every Markov process is a martingale, as discussed in this question.