Electronic – Common Emitter amplifier is used as

In a common emitter amplifier with an emitter resistor the collector current and the emitter current are almost identical.

"Gain" is proportional to collector voltage / emitter voltage (by definition) so as Vc = Ic x Rc and as Ve = Ie x Rc and as Ic ~= Ie, Gain = Rc/Re.
Rc = collectpr resistor and Re = EFFECTIVE emitter resistance. This is the transistor internal Re_internal + theemitter resistor.

If there is no emitter capacitor across the emitter resistor then
Gain = Rc / (Re + Re_internal).
If the emitter resistor has a capacitor across it which allows AC to pass with no loss (or not much loss) then
Gain ~= Rc/Re_internal.

Re external is used to set the DC operating conditions and maximum AC gain occurs when the only emitter resistance = Re_internal.

For a silicon transistor Re_internal "just happens" to be 26/I ohms where I is the emmitter current in mA. This is a basic property of the silicon junction which I'll not go into further here. It makes sense when examined but is not intuitive.

ie if i=1 mA then Re_internal = 26 Ohms.
If Ie = 4 mA the Re_internal = 26/4 = 6.5 Ohms
If Ie = 10 mA then Re_internal = 2.6 Ohms

SO for a fully AC-bypassed Re (large enough Ce) :

• Gain = Rc / Re_internal = Rc / (26/mA)

• Following this through you arrive at the "fact" that
  Gain = 38.4 x Vc
  Not many people know that.
  Few people believe it when told :-)

There are a number of things wrong here

1. There are no component designators in the schematic. This makes it hard to talk about, so I'll be more brief than otherwise.

2. You should use junction dots at all points where nets connect. The four lines crossing left of the transistor is ambiguous.

3. A quick calculation of the bias point not taking base current into account results in 1.3 mA thru the transistor, so your claim of 1.26 mA is plausible. However, this should tell you there is no way you're going to get anywhere near 1 W of output power. Even looking at the transistor as a perfect switch, the collector swing will be from 6.6 to 24 volts, for a swing of 17.4 volts. However all this is at a impedance of 8 kΩ. 17.4 Vpp sine wave is 6.15 V RMS. At the best case power transfer, the load is 8 kΩ so the voltage is half that, or 3.1 V RMS into 8 kΩ, which comes out to 1.2 mW. That's less power (although at a different voltaqe) than the MP3 player is likely putting out.

4. The frequency response is a mess. The gain of the amp only 2.7 at low frequencies. This starts going up 3 dB/octave at about 50 Hz, but the high end is unpredictable because it relies on properties of the transistor you don't know. Then the cap in series with the speaker forms another high pass filter, but this also allows the speaker to load the output heavily probably having a reverse effect for part of the range.

5. I can't even begin to guess what the 12 kΩ resistor accross the speaker is supposed to do. It makes no sense whatsoever.