Can we have physical quantities which have magnitude and direction but are not vectors? I am not able to understand how to approach the question. 
Vectors are defined as quantities having magnitude and direction, then how is it possible?
Please explain.
 A: Yes. Electric current is an example. It has a direction and magnitude but it doesn't follow vector summation rule, so it's not a vector. However current density is a vector.
A: I am guessing that you are encountering vectors for the first time. Vectors have been presented as things with magnitude and direction, like little arrows. You are asking for clarification on what vectors are all about. 
Little arrows are a good example of vectors. But yes there are vectors that do not have magnitude and direction. 
To a mathematician, vectors are things that can be be added together and multiplied by numbers. You add arrows tip to tail. The result is an arrow. When you multiply an arrow by 2, the result is an arrow twice as long. The complete definition is technical, but this is the idea. 
Functions of the form $y = ax + b$ fit the definition. You can add two such functions together and multiply a function by a number. You can invent ways to give them a magnitude, such as area under the curve when you graph them from 0 to 1. But hey have no direction. 
Notice that numbers fit the mathematical definition of vectors. The have magnitude, but no direction. 

Physicists add another requirement to what they consider to be a vector. It must transform in the right way. 
Suppose you have an arrow pointed straight ahead. You also have a number, 3. You rotate a quarter turn to the left. Now you see a vector of the same length pointed to the right, but the number is no different. This is a trivial example, but the idea becomes important for relativity. 
Suppose you have a stationary charge. The charge is surrounded by an electric field. In elementary physics this is treated as a vector much like little arrows. If you have many charges, the fields add and multiply by numbers in the right way. If you turn to the left, the field changes direction, but stays the same magnitude. 
But Einstein showed that electric field and magnetic fields are better considered a single more complicated thing. If you run by the charge, you see both electric and magnetic fields. A vector should not change in this way when you run past it. 
The electromagnetic field is an example of a tensor. You could say it has two magnitudes and directions.  
A: yes, electric current have both magnitude and direction but it is not a vector because the formula of e.c. is V/R and voltage and resistance are scalar quantities .and two scalar quantities can never give a vector quantity .and it follows simple algebra rules.
A: Tensors have angles and magnitudes but are more complicated than vectors. One may have a quantity with a direction and magnitude  which will not follow vector rules because a tensor is defined with 9 components, not three. It has the square of the dimensions of the vector space, here 3 dimensions, but one can have n order vectors.
The most recent physical tensor I have seen are the E and B fields in the BICEP2 experiment, an attempt to get  the imprint of the tensor polarization from the gravitational waves 


Left: BICEP2 apodized E-mode and B-mode maps filtered to 50 < ℓ < 120. Right: The equivalent maps for the first of the lensed-ΛCDM+noise simulations. The color scale displays the E-mode scalar and B-mode pseudoscalar patterns while the lines display the equivalent magnitude and orientation of linear polarization. Note that excess B mode is detected over lensing+noise with high signal-to-noise ratio in the map (s/n > 2 per map mode at ℓ ≈ 70). (Also note that the E-mode and B-mode maps use different color and length scales.) 

The dependence is complicated but directions and angles in the physical measurement can be used to extrapolate to a tensorial input.
A: No, quantities having magnitude and direction should not be necessarily to be a vector quantities because it must obey the law of vector addition. For example Electrical current
