I am reading through power amplifiers, at a first glance I though it is like voltage amplifier in which applied voltage signal is amplified. Likewise, A power amplifier "should" be an amplifier whose imput power level is amplified to get a higher output power.now my question is is the term "power" relative to a particular load? For example Let's say a speaker wants 1000Watts of power having resistance of 10 ohms.it means it's required voltage is 10V. An amplifier which can take 1 volts input and makes it 10volts and which has the capability to drive this speaker can be said as a "power amplifier " w rt to the speaker. Secondly we are considering another amplifier which can take 1 volts if input and 100 volts of output, but this time it's output power say, 20 w , this time voltage gain is 100, much more than the first one, but it's power is lesser, so it can't be a power amplifier, am I right?
A power amplifier provides more output power than it requires at its signal input, even if the output voltage is not greater than the input voltage. Full stop.
Your examples are very confusing, as you never say anything about input power, just input voltage. You should also check your math regarding a 1000 W output into a 10 ohm load.
As you suggested, power is defined with respect to some load. Your math is a little incorrect though. If I have a 10ohm speaker and I want to pump 1kW of power into it, I need 100Vrms across. Remember that power is P=VI=V^2/R=I^2*R. (In terms of rms voltages and currents. If you use sinusoidal amplitude, you need to divide your voltage and current by the square root of 2).
So that being said, the output power of an amplifier depends on what load it is driving. For example, when an audio power amplifier is specified at some power level, it is always with respect to some speaker impedance (e.g. 8 ohms). Likewise, the input power depends on what source resistance is driving the amplifier. At which point, we can begin to talk about power gain.
A part you seem to be missing is the current gain of an amplifier. I can have amplifier with voltage gain less than one (e.g. a common collector amplifier) but still have a power gain greater than one. This is because power gain is current gain times voltage gain, and some amplifiers can provide current gain without much voltage gain.
Now the difference between an ordinary amplifier an a power amplifier is that power amplifiers tend to be driven large signal (high voltage and current swings at the same time). Since transistors are nonlinear devices, this large output swing results in the characteristics of the transistor bias to change over a full signal period. These nonlinear effects generally introduce distortion into the output that is almost always undesirable. Since normal amplifiers are typically driven in a small signal regime (small input, relatively less small output), these nonlinear effects are not as important. You might also see power amplifiers that have far less voltage gain than other amplifiers. This is because these amplifiers provide high current gain instead.
Also, the efficiency of a power amplifier is usually far more important than a small signal amplifier. There's multiple definitions for power amplifier efficiency, but one of them is the total signal output power over the consumed DC power. The lower the efficiency, the more power is wasted as heat, meaning you need more heatsinking and beefier power supplies. Heat tends to be the biggest problem in really high power amplifiers.
To underline this efficiency problem, you may see special configurations for power amplifiers in order to improve efficiency. For example, a class AB push-pull power amplifier has a much higher efficiency than a class A power amplifier at the cost of increased distortion. It's always a trade-off.