Band-gap is the forbidden energy range, in which no states are allowed, i.e. the density of states in the band-gap is zero. There are many such forbidden energy ranges in the semiconductor band structure; however, we generally refer to the forbidden energy range between the conduction band and the valence band, in other wards, the energy difference between the conduction band minimum and the valence band maximum. 

The earlier answers are OK for the band-gap, but, is incorrect regarding the Fermi level. 

There is a lot of misunderstanding regarding the Fermi level. What we refer to as the Fermi level is actually the chemical potential, as defined in the Fermi-Dirac distribution or probability function. Often the Fermi level is confused with the Fermi energy, as has been done in the above answers. Fermi energy, as has been correctly stated in the above answers, is the energy of the highest occupied state at absolute zero. Fermi energy has no meaning at any temperature other than 0 K. But, the chemical potential (Fermi level) exists at all temperatures. The Fermi energy for a semiconductor is the valence band maximum, as that is the highest filled state at 0 K. The Fermi level for a semiconductor at any temperature (and doping) is determined by the charge neutrality condition, i.e. the positive charges equal the negative charges in magnitude. The charges (holes, ionised donors, electrons, and ionised acceptors) are all determined by the Fermi-Dirac probability, in which the chemical potential (Fermi level) is the most important parameter. 

No,

Fermi level is highest occupied energy level at zero temperature while band gap isthe difference between valence band and conduction band. 

thanks

No. 

Band gap is an energy difference between maximum of valence (HOMO) level to the minimum of conduction (LUMO) band level,

whereas, fermi level is the maximum energy state of electrons occupied in the material at absolute zero temperature, most of the time it falls inside the band gap for semi conductors, hence at room temperature itself certain carriers avail in conduction band of semi conductors. 

Dear Maninder

I agree with the above. The band gap of a semiconductor is the energy difference between the valence band and the conduction band edges of inorganic materials (or between the HOMO and LUMO of an organic semiconductor). You can estimate the (optical) bandgap of a semiconductor by using optical absorption measurements (from the absorption edge or, more accurately, from the tauc plot). The Fermi level is an energetic level usually located within the band gap of the material which represents the maximum energy occupied with electrons at absolute zero temperature. The position of the Fermi level is a criterion for the type of the semiconductor e.g., when it is near the conduction band edge you have a n-type semiconductor. You can estimate the position of the  Fermi level by measuring the work function which is the energy difference between the fermi and the vacuum level. To measure the work function you need to perform ultra violet photoemission spectroscopy measurements.

Maria has answered the concerned question perfectly.

Band-gap is the forbidden energy range, in which no states are allowed, i.e. the density of states in the band-gap is zero. There are many such forbidden energy ranges in the semiconductor band structure; however, we generally refer to the forbidden energy range between the conduction band and the valence band, in other wards, the energy difference between the conduction band minimum and the valence band maximum. 

The earlier answers are OK for the band-gap, but, is incorrect regarding the Fermi level. 

There is a lot of misunderstanding regarding the Fermi level. What we refer to as the Fermi level is actually the chemical potential, as defined in the Fermi-Dirac distribution or probability function. Often the Fermi level is confused with the Fermi energy, as has been done in the above answers. Fermi energy, as has been correctly stated in the above answers, is the energy of the highest occupied state at absolute zero. Fermi energy has no meaning at any temperature other than 0 K. But, the chemical potential (Fermi level) exists at all temperatures. The Fermi energy for a semiconductor is the valence band maximum, as that is the highest filled state at 0 K. The Fermi level for a semiconductor at any temperature (and doping) is determined by the charge neutrality condition, i.e. the positive charges equal the negative charges in magnitude. The charges (holes, ionised donors, electrons, and ionised acceptors) are all determined by the Fermi-Dirac probability, in which the chemical potential (Fermi level) is the most important parameter. 

Fermi level and band gap are not same. These are two different parameters.Fermi level in semiconductors can be moved up or down from the center of gap by increasing the density of electrons or holes without changing the band gap. This happens in n type or p type semiconductors. If extra charge carriers electrons and holes both are created in equal number in an intrinsic semiconductor, then Fermi level splits in two quasi Fermi levels, one for electrons which moves up and other for holes which moves down.This happens in case of photoconductivity in presence of light. In conclusion, one can say that the position of Fermi level is dependent on the concentration of majority charge carries even though  the band gap remains the same. Band gap is determined by optical methods while the position of Fermi level from conduction or valence band can be determined from temperature dependence of conductivity.

Dear Maninder,

Fermi level nd enrgy bandgap are quite diffferent. In fact, a material need not to have a bandgap to have Fermi level. As you know, Electrons follow Fermi-Dirac statistics and accordingly Fermi level is that energy level for which the probability of electron occupancy is 50%. While semiconducting materials have energy bandgap between conduction and valence band. Fermi level in such material is in middle of this bandgap.

Fermi energy is the maximum energy possessed by the charge carriers. Just for an illustration draw spheres of different radius, the radius of which corresponding to velocity vectors of various charge carriers : then the sphere drawn with largest velocity will corresponds to fermi energy. Band gap is the energy difference between the maxima of valance band and  minima  of conduction band. However, Fermi level is decided by the concentration of charge carriers in semiconductors which may have any position in between band gap.

No,

See Maria

Maninder, 

consider the answers given by Prof. Samares Kar and Maria they have justified your question with valuable explanation, It really needs deep understanding. To all Thanks for the efforts.

Prof. Samares Kar, I cleared myself with your explanation quoting your two points

1. Fermi energy has no meaning at any temperature other than 0 K.  But, the chemical potential (Fermi level) exists at all temperatures.

2. The Fermi energy for a semiconductor is the valence band maximum, as that is the highest filled state at 0 K. 

could you please shine some light on the following query.

Q: At 0K we have conduction band (where empty energy levels exists) and filled band is the valence band (Highest occupied state corresponds to Fermi energy thus fermi level at 0K). So how do we define valence band at a temperature other than 0K, why one would bother about fermi level -where there are no allowed states at all?what actually chemical potential is and tell?and how this defined chemical potential (fermi level) is significant where there are no states at all?

Thanks in advance.

@Sateesh Prathapani@

Answer to "what actually chemical potential is and tell?and how this defined chemical potential (fermi level) is significant where there are no states at all?"

Fermi Level is the indicator of (or reference for) Fermi Dirac distribution (probability of finding carriers). Fermi Dirac Distribution doesn't tell you the number of carriers or energetic location of carriers directly which is the useful number for understanding device working. It is just the probability distribution (i.e. carriers might present or might not present) which is represented by the Fermi level where the probability is 50% (for any temperature). Fermi level might present in the bandgap where no states are available. Remember, we have to map the Fermi Dirac distribution with Fermi level is used as a reference line. The overlap of available density of states and Fermi Dirac Distribution will provide the carrier density (as shown in attached file which is taken from Pierret, Semiconductor device fundamentals).

So, if Fermi level is located where no states are available - It doesn't matter. But, if there is overlap of fermi dirac distribution (drawn/visualized using Fermi level as reference) and DOS, it provides lot of useful information.

I request the readers to correct for any mistakes in the understanding.

Fermi level is located, above which no states are available (Highest occupied energy state)