Safety warning This page provides conceptual schematics of various types of push-pull power inverters. The simplest one is a free-running converter called Royer oscillator. It was patented by Bright and Royer in 1954 (I don't know why this circuit does not bear the name of the second co-author). An example of the parts list: Q1, Q2- 2N6487 (or ZTX849 for low wattage applications); R1=4.7 kOhm, R2=470 Ohm. Here is how it works. When power is first applied, it forward biases both transistors. If both of them will be in on-state, they will just short both the transformer and the input source. However, practically, Q1 and Q2 will not be "on" at the same time because their characteristics are never exactly the same and their base drive windings are out of phase.

Royer-based inverter schematic In practice, one of them (say, Q1) will turn on more and some voltage will appear across transformer's primary winding. This in turn induces voltages in the base windings of such polarities that they drive Q1 further into saturation and Q2 into cut-off state. As current flows through Q1 and the half of the primary, magnetic flux in the core increases linearly with time. At some point it will approach saturation when the flux can no further increase. The voltages across all windings will drop to zero and then reverse polarities. This will cause Q2 to conduct and Q1 to be in cut-off condition. This self-oscillating process will continue and a bipolar rectangular voltage will be generated across the output. Its frequency depends on the time required for the core to saturate. We can derive from Faraday's law that Vin=4N1FBAc10-8, where Ac- core's cross-sectional area in sq.cm, B- saturation flux in gauss. Therefore, for a desired frequency F, you need to select the transformer with N1Ac=Vin108/4FB. Then secondary turns will be selected based on desired output: N2=N1Vout/Vin.The main drawbacks of this circuit are unstable frequency, square wave output, lack of voltage regulation (which can be done only by varying Vin) and lack of current limit. It can be used to power bulbs and other resistive loads that are not sensitive to frequency or waveform.

Current-fed inverter circuit A modification of Royer oscillator has an additional inductor L in series with Vin and a capacitor C between two collectors. With this configuration the resulting waveform is sine-like and the oscillating frequency is determined by resonance of magnetizing inductance L1 and C: F=1/(2πsqrt(L1C)). For example, L1=400 uH and C=0.1 uF would yield F=25 kHz. This is a conceptual schematic of such current-fed oscillator controlled by a buck regulator. The MOSFET in the buck can be driven from an external PWM to regulate the output level. This topology is widely used to feed cold cathode fluorescent lamps (CCFL) with high-voltage of 20 to 100 kHz (for practical schematics see for example this application note). It is not quite suitable for 120V 60 Hz applications, where a sinusoid is normally produced by using PWM technique.
The circuits with bipolar transistors are not very efficient because you need to continuosly provide substantial base currents. The diagram below shows a more efficient current-fed inverter implemented with MOSFETs. Resistors R1, R2 provide turn on, while the diodes D2, D3 provide turn off and isolate the gates from high voltage on the drains.
MOSFET inverter Here is an example of the parts list: Q1, Q2, Q3=STP36NF06L, R1, R2=10 Ohm, D1=MBR1035, D2, D3=MUR120, L=100 uH, C=0.1 uF. The current-fed concept can be especially helpful if you drive the switches from an external circuit. Such approach is often used to maintain a fixed frequency independent on the cores properties. The drive frequency has to be below the self-resonance though. Note since volt-seconds of each half of cycle will never be exactly the same, without the inductor, the transformer in push-pull converter with external excitation can suffer from DC component of the magnetic flux. Eventually this imbalance can cause saturation of the core and failure of a transistor. There are plenty of hobbyist schematics floating around the web that ignore this possibility and do not have the inductor.
Also see our tutorial to DC to AC conversion and this guide to solar inverter operation.