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Real op-amps are far from perfect, one of the biggest problems most people run in to is input offset voltage, this essentially acts like a small DC voltage in series with your input, this is particularly troublesome for high gain DC amplifiers where even a tiny offset can cause significant error, this can be reduced in a number of ways, one way is to use a more expensive precision op-Amp, another common solution is to null the offset after the op-amp with a trimmer, some op-amp even have dedicated offset null pins like the classic 741, the only way to truly eliminate input offset voltage is the use of a chopper stabilized op-amp although these do come with their own drawbacks.
Input offset current is another source of error, usually only a concern when dealing with high impedance (resistance) circuits, real op-amps draw a small amount of current through their inputs, usually in the nA to uA range, this may not sound like much but if you have a large resistance of say 1meg a significant voltage drop can occur, op-amps are available with less than 1nA input offset current for high impedance applications.
As a general rule pick bipolar for low offset voltage and FET or better MOSFET for low input offset current, one thing to keep in mind is offset can change with both age and temperature, in some cases by an order of magnitude or more so always check the datasheet.
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