Summarised_Solution:
Description provided of fundamental action of bipolar and unipolar stepper motors and basic pros and cons of each.
Comment on supplied unipolar driver circuit.
Similarity to already suggested bipolar driver circuit shown.
Vijay proposed the circuit below as a possible starting point.
This is a circuit for a unipolar stepper motor controller, whereas Vijay requires a bipolar controller.
In a bipolar stepper motor there are (usually) two coils and these are driven with either of two possible polarities at a given moment. The effect is to advance the stepper over a 4 step cycle. If polarities are taken as forward or reverse and the two coils are name A and B then the stepping seuqence is -
A forward
B forward
A reverse
B reverse
A forward ... (repeating)
A unipolar stepper motor controller uses (usually) two coils both of which are centre tapped. The centre taps are connected to (usually) positive and alternate sides of a given coil are grounded to activate the coil. As the two halves of a coil are magnetically coupled, activating the left side of a coil has the same effect as if the right side of the coil had been activate with reverse polarity. The same result is thus achieved by using A-left, A-right as would be achieved by A-forward, A-reverse with a bipolar stepper motor.
A bipolar steppe motor has the advantage of activating the whole winding at a time so the copper and iron involved are better utilised than in a unipolar stepper motor where half of a coil's copper and iron is "wasted" when the other half is activated.
A unipolar stepper motor has the advantage of only requiring a single switch to ground (usually) per activation rather that the full H bridge switch with one switch "up" and one switch "down" required by a bipolar stepper motor.
A eldom realised side effect with a uni-polar stepper motor is that when a centre tap is connected to V+ and eg A-left is grounded, A-right will rise to 2 x V+ due to transformer action in winding A. This is no great problem as long as the off driver on A-right can withstand 2 x V+ and attempts are not made to clam winding ends to V+ - as is quite often done in circuits shown on the internet.
SO:
This is a unipolar stepper motor driver.
The BC547 transistors are successively switch to ground to enable half of each winding as required, as described above. To convert this circuit to bipolar operation it would be necessary to provide 4 x high side transistors and drive them correctly This could be done but requires subtantial extra circuitry and also level shifting for the high side drivers.

This is essentially what is done in this circuit (mentioned in my other answer).

Below is a single winding driver at larger size.
The upper and lower FETs act as switches to V+ or ground and the BC547 transistors act as level shifters. The 4013 flipflops act as sequence controllers as before. This is an OK circuit but not the simplest that could be built - see my other answer.

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