I am using a standard 741 Op Amp from TI. For me to amplify any signal, I am required to set +Vs = 15 Volts and -Vs = -15 Volts. I am now concerned as to what will happen if I connect -Vs to 0 (ground) instead of-15Volts. Will the Op Amp work as it does normally ? How is the gain affected ? 
Answer
There are two problems with having the -Vs of a 741 connected to ground.
First, as others have said, the output cannot go below ground. In fact, with a 741 it can't even get very close to ground, only to within maybe 1V or 1.5V of ground. This is called the output swing and is usually rated with some kind of a load resistor. Take care that the load resistor configuration for the ratings represents your setup (for example, an LM324 may be rated with a 10K to -Vs, but if you connect a 10K to +Vs it will no longer be able swing as close to ground (there's an internal 50uA pulldown).
Secondly, the inputs won't work if they are too close to -Vs (or ground in this case). Again, it's about a volt or a volt and half with respect to -Vs. So even if you put a zener diode or something like that in series with the output to allow it to reach ground, you could still not make, say, a unity gain buffer that would work below that voltage (called the common mode input voltage range).
For a good design, the inputs should never exceed the guaranteed input common mode range, and the output should never exceed the guaranteed swing if you expect the amplifier to behave in a fairly linear manner. We do break those rules at times, but there should be a sign "dragons be here" because many strange things can happen- phase reversal, exceedingly long delays recovering from being 'railed', surprisingly high input current and so on..
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