What is an Ideal Op Amp?
An ideal op amp is an op amp that has perfect conditions to allow it to function as an op amp with 100% efficiency.
An ideal op amp will display the following characeristics, of which are all explained in detail below. Ideal op amps will have infinite voltage gain, infinitely high impedance, zero output impedance, its gain is independent of input frequency, it has zero voltage offset, its output can swing positive or negative to the same voltages as the supply rails, and its output swings instantly to the correct value.
In real life, as with all ideal components, an ideal op amp does not exist. However, if we can get an op amp to display as close as possible the characteristics of an ideal op amp as closely, we can make a more efficient op amp that has better output in realworld conditions.
Now we will go over all of these characteristics of an ideal op amp mentioned above, so that you can know what each is and so the difference between an ideal case and a real case.
Below is a table that charts each of the major characteristics of an op amp and how they differ in ideal and real op amps.
Ideal Op Amp Characteristics
Characteristics  Ideal Op Amp  Real Op Amp 
Infinite Voltage Gain  An ideal op amp will have infinite voltage gain.
Op amps are devices that many times are used to function as amplifiers.
A voltage is input into the op amp and as output, it produces the voltage amplified. An ideal op amp will produce megagain, practically, it will be able to produce infinite gain. It will amplify the signal infinite times over so that we can have as much gain as we'd ever need. 
A real op amp can only produce a finite gain. 
Infinitely high input impedance  An ideal op amp will have infintely high input impedance. This will ensure that the op amp causes no loading in the circuit. The lower the input impedance, the more current that an op amp will draw. The higher the impedance, the lower the current that an op will draw. We want high input impedance so that the op amp doesn't disturb the original circuit by pulling current from it. To do this, we need infinitely high input impedance.  A real op amp has finite input impedance. Even though many types of op amps, such as MOSFETs, have extremely high input impedance, in the order of teraohms, it is still finite. 
Zero Output Impedance  An ideal op amp will have zero output impedance. When an op amp produces its output signal, we want the op amp to have zero voltage so that the maximum voltage will be transferred to the output load. Voltage is divided in a circuit according to the amount of impedance present in a circuit. Voltage drops across a component of higher impedance. In order for the voltage to drop across the output load, that load must be of greater impedance than the output of the op amp. This is why, ideally, we want the output impedance of the op amp to be zero  A real op amp will always have some output
impedance, though it is low. A typical value can be 75Ω.

Gain Independent of Frequency  In an ideal op amp, the gain that the op amp produces will be independent of frequency. This means that regardless of the frequency of the input signal going into the op amp, the gain that is produced will be constant and good across all frequencies.  In real op amps, the gain that is produced is only for a certain bandwidth of frequencies. Outside of this bandwidth, the gain that the op amp produces will decline. 
Zero Input Voltage Offset  In an ideal op amp, if no voltage is applied to the inverting and noninverting input pins, the op amp will output a voltage of 0, since there is no difference at all of the voltage applied to the 2 input pins.  A real op amp will have slight offset even if the voltage applied to the pins are the same. To correct this offset, voltage must be applied to the offset pin. 
Positive and Negative Voltage Swings to Supply Rails  In an ideal op amp, the ac voltage which is fed into the op amp to be amplified will swing all the way up for the DC positive supply rail and all the way down for the DC negative supply rail, making 100% efficient use of the DC voltage supplied to an op amp.  In real op amps, the amplified signal will not fully reach the DC supply rails. They will fall short of it. 
Output swings instantly to the correct value  In an ideal op amp, the output will swing instantly to the amplified voltage value. There will be no time delay between the time the voltage is input into the op amp till the time it is output. It will all be instantaneous.  In real op amps, the amplified signal will take time to reach the fully amplified voltage value. This is determined by the slew rate of the op amp. 
Again, ideal op amps can't exist because op amps, as all electronic components, will have some internal resistance, which won't allow maximum efficiency. However, if we can get
real op amps as closely as possible to ideal conditions, we will have very efficient, useful op amps.
Related Resources
Why Does an Op Amp Need a High Input Impedance and a Low Output Impedance?
Why Does an Op Amp Need a Power Supply?
How to Connect a LM741 Op Amp Chip to a Circuit
Voltage Follower
Unity Gain Buffer
Isolation Amplifier