In most sources, the Brownian Motion and the Wienner Process are the same things. However, in some sources the Wiener process is the standard Brownian motion while a general Brownian Motion is of a form αW(t) + β.

A Brownian  Motion or Wienner process, is both a Markov process and a martingale. These two properties are very different. In fact, they have little in common. A random process X_t, adapted to a filtration F_t, is a martingale with respect to the filtration if conditional expectation of the increment E( X_t - X_s | F_s) =0 for all t>s (conditional increment equals 0).

The same process has Markov property if 

E( X_t  | F_s) = E( X_t | X_s) =0 for all t>s (future distribution is independent of past).

 martinagle = fair game, markov= memoryless.

The wiener process is often called standard Brownian motion

if a process is a martingale, then the future expected value is dependent on the current value of the process while in markov chain the probability of future value (not the expected value) is dependent on the current value only. Are the following true? 1. Martingale is a subset of markov processes because there can be many markov processes whose expected future value is not equal to the current value. 2. If martingale is strictly a markov process then the only difference is that in a markov process we relate the future probability of a value to past observations while in a martingale we relate the future expected value given all past observations.

Reference https://www.physicsforums.com/threads/difference-between-martingale-and-markov-chain.477725/

The Brownian motion and the Winner process are the same thing.

The Brownian motion is such a good process that it is both a Markov process and a martingale.

The mean value of a martingale is always the same, but the mean value of a Markov process may not be the same. For example, Brownian motion with a constant drift is a Markov process but is not a martingale.

What I understand now is:

1) Brownian Motion is a concept in Physics and its mathematical explanation is known as Wiener Process.

2) Markov and Martingale are two properties of Wiener process. Either they both exist in Wiener Process or only one state i.e. Markov.

3) In Markov the transition Matrix among different states remain same over the process. Based on the probabilities of transition of one state to other, the future probabilities of transition of the present state into future states can be calculated.

4) The Wiener Process without drift is Martingale. Which means that the mean value (Expected value) of the process shall remain same as of the current value of the process. However, the Wiener Process with drift is not a Martingale.

What else should I know??

Thanks to @Zeev Sobol , @Rezvan Kamali  and @Chen Jia  for help.

Have you heard anything about Brownian noise ??

White noise has mean 0 and std dev of 1. In other words a random variable. Brownian motion is when you take the running sum of these random variables. 

In order to identify noise amidst actual signals we often employ the power spectrum. We can calculate the power spectrum by taking the square of the time series fourier transform. 

The power spectrum will give a constant power with respect to frequency in the case of white noise. 

If we take the power spectrum of Brownian motion you will have power that slopes as a function frequency at the rate of f -2, hence it called Brownian noise. 

Depending on the steepness of the power spectrum with respect to frequency, we also have pink, black and red noise among others. 

Edit: I made a mistake previously. I confused wiener process with white noise. Corrections has been made to reflect the changes. Apologies.

In most references, Brownian motion and Wiener process are the same. In fact the Brownian motion is a continuous process constructed on a probability space, nul at zero, with independant increments such that the increment Bt - Bs   has Gaussian distribution with mean 0 and variance (t - s). The Wiener process is the Brownian motion constructed by Norbert Wiener on the canonical space ie: the space of continuous functions, equipped with the filtration of coordinates, on which a measure has been constructed, called the Wiener measure. The whole collection is called the Wiener process.  It is clear that the Wiener process and any Brownian motion constructed on a different probability space have the same distribution, called the Wiener measure.     

The martingale property is fundamentally different from the Markov property. The martingale property is crucial in stochastic integration while the Markov property is important for semi groups and Kolmogorov PDEs linked to diffusion processes. 

Martingales are NOT a subset of Markov processes. Here is a reference how you may construct a martingale which is not Markov.

https://djalil.chafai.net/blog/2012/01/20/martingales-which-are-not-markov-chains/

The simplest example is: be Z_n, n=1,2,3,...., a sequence of i.i.d. random variables centered at zero, and X_0 be a bounded non-zero random variable independent of Z_n, define X_n=Z_nX_0 + X_{n-1}, n=1,2,3,.. Then X_n, n=0,1,2,.., is a martingale, but not a Markov process.

A Brownian motion with a drift is an example of a Markov process which is not a martingale.

"In Markov the transition Matrix among different states remain same over the process" - no, it can change in time. Then you will get a non-autonomous Markov process.