In high school we have always been taught that wave looks figure $A$ and from there we have learnt amplitude,frequency etc of waves. But in a lot of books, waves are represented by the second figure and are often told waves look like this. I know that the circumstances are wavefronts here but they are not waves. That's why i can't think of wavefront and wave as the same. How can i think of these $2$ samely?
Consider throwing a stone in a pond. From the point where it hits the surface, circular waves propagate radially outwards. Now imagine looking at it from the top and marking all the "hills" or all the "troughs" – you end up with something like your Figure B.
Now imagine looking at the surface of the water from one side, like a vertical slice of it. The surface will look like your Figure A.
Both are correct, they are just different perspectives of the same thing.
Briefly, definition of wavefront is the set of the points that have same phase. Hence, wavefront in your figure 2 is just connection of creast(hill) or trough(valley) of wave. It is anologous with equipotential concept.
Wave function in 2D (or 3D) is multi-variables function, so such wavefront description is trivial. Also, it has benefit to understand behavior of waves easily. (like as plane or spherical wave)