Fleming's right hand and left hand rule Why are there two rules: Fleming's left hand and right hand rules?  What is the difference between the two and why can't we use just one rule? 
Suppose the magnetic field is from right to left and the motion of the wire is downwards then according to the right hand rule the induced current will be in the straight direction.
But if we use the left hand rule in this same situation to find the direction of motion of wire then it shows that the direction of wire is upwards.
Please help me.
 A: Similarities: in both the rules the thumb gives the direction of force/motion, the index finger gives the direction magnetic field and the middle finger gives the direction of current.
Differences:
1) Left hand rule: This rule is used when magnetic field direction and current direction are given and you have to find the direction of force/motion of the conductor.
2) Right hand rule: this rule is used when the magnetic field and force/motion of the conductor is given and you have to find the direction of the current.
In both the rules all three should be perpendicular to each other.
A: It is unfortunate that the physics of magnetism got saddled with several different *-hand rules, and that they use different hands. Let's pull them apart:
Fleming's left-hand rule
gives you the direction of the force that acts on a current if you know the magnetic field.

Image source
This rule applies to motors, i.e. devices which use currents in a magnetic field to generate motion. It derives its validity from the Lorentz force,
$$
\mathbf F=q\mathbf v\times\mathbf B,
$$
in which the current goes with the charge's velocity and the induced motion is along the direction of the force. This is why this rule coincides with the left-hand rule used in cross-products in general.

Fleming's right-hand rule
is much less used in physics (though I can't speak for how engineers do things). It applies to generators, i.e. devices which use motion in a magnetic field to generate currents. This again relies on the cross product in the Lorentz force, except that now the charge's velocity is given by the object's motion, and the force along the wire is what establishes the current. This means you've swapped the middle finger with the thumb with respect to Fleming's left-hand rule, which you can do by keeping the (vague) assignments to 'motion' and 'current' and switching hands.

Image source
I dislike this convention very much and I would encourage you to forget all about it except the fact that it exists and should be avoided. In any situation where you need it, you can simply use the Lorentz force to figure out which way the current will go.

Ampère's right-hand rule
is quite different, and it gives you the magnetic field generated by a straight wire.

Image source
It derives its validity from the Biot-Savart law, which gives the magnetic field at position $\mathbf r$ generated by an infinitesimal current element of current $I$ and directed length $\mathrm d\mathbf l$ at position $\mathbf r'$, as
$$
\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\frac{I\mathrm d\mathbf l\times(\mathbf r-\mathbf r')}{|\mathbf r-\mathbf r'|^3}
$$
Again, it is the cross product which dictates the direction of the field, and you should check by yourself that it works out as indicated in the picture.

As you can see, the rules are quite different. It is therefore crucial that, if you want to use them as mnemonics, you learn correctly which one applies where, and that you apply them correctly. (It is no use to learn which hand to use if you e.g. swap the assignments for the index and middle finger.)
The most important thing to learn, though, is the Lorentz force law, which is based on a left-hand rule (charge-times-current on your middle finger, field on the index, force on the thumb) indicated by the cross product. This is essentially failsafe if you apply it correctly and is less subject to confusion with other rules.
