How a sin wave can form in one dimension? One dimension have no up down or side, just a straight line path. But when we are reading about one dimensional wave, it moves up and down if it's going along x axis. How its possible?

 A: If this is a wave on a piece of string then yes it does require (at least) two dimensions. That's because the amplitude of the wave is its displacement in the direction normal to the string. But there are lots of waves where the amplitude is not a physical displacement in a dimension, so these waves do not need a second direction.
For example consider a light wave. The amplitude of a light wave at a point is the value of the electromagnetic field at that point. For example for a one dimensional light wave travelling along the x axis the electric field of the light would be given by:
$$ E(t,x) = E_0 \sin(\omega t - kx) $$
This wave doesn't need a second dimension because unlike the wave on a string it isn't oscillating in a physical dimension.
For completeness we should mention that even for mechanical waves, like a wave on a string, the motion in the second dimension is small compared to the distance travelled. In that case the wave is approximately one dimensional. Also note that as my2cts mentions there are waves like sound waves that are longitudinal i.e. the oscillation is along the direction of travel. So these waves need no second dimension. And finally, of course, the real world is three dimensional so one dimensional waves are a mathematical abstraction and we shouldn't be too worried that they don't really exist.
A: You can still have a longitudinal wave of varying compression and expansion along x. 
A: I understand one dimensional wave equation, represented by sin and cos as:
1) It is a plane wave, so we don't have to worry about the actual situation of losing energy with distance (as it happens when a stone is dropped in a lake). It is a good approximation far from the source, and if the observed length is not to large. 
2) We choose our axis so that one of them (usually x) is the wave velocity direction.   
The transverse vibration is not a problem because the simplification of (1) and (2) is the meaning of the one dimensionality approach.
